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Assume a planet that always presents one side to the sun. No moons. The orbit of the planet around the star is essentially spherical, very minor and undetectable aberrations. The orbital period of the planet around the star is in multiples of the lifetimes of the sentient beings. That is, no sentient being lives long enough to go through one complete rotation.

NOTE: this question is not about HOW this would happen, it is to be taken as a given that it DID happen.

A sentient species evolves on this planet.

There are no day/night cycles, no seasons, and no cyclical changes in the sky - it would be constantly different throughout one's lifespan. No repeats. Life evolved without any natural circadian rhythm. Plants grew and died independently of any cycle. No defined growing season. A plant could sprout at any time, and die at any time. No menstrual cycles, no estrus cycles in animals. There would be no biological basis on which to establish any. Life would have evolved without any concept of cyclical time passage. As far as life was concerned, everything was eternally homogenous, time wise. Nothing happened in unison.

To these sentient beings, the passage of time is irrelevant. They have no way to compare the passage of time between people. Heart beats are different, respiration is different, one's pace in walking is different. The interval to travel from one point to another is different. It is human experience that what seems like a long period of time to one person is a short period of time for another person. The beings get to where they are going when they get there, without reference to anyone else's travels. They eat when they need to, irrespective of how long it has taken between meals (think of a snake, that can go for very long periods between meals, depending on how big the last meal was). Gestations are never the same length, so even if two beings got pregnant at the same time, the deliveries would never likely be at the same time. Two objects released from a height at the same moment would be observed to arrive at the ground at the same moment, but there would be no indication that if dropped at some other moment they would arrive at the ground in the same time as these two, without some form of quantized time.

Many of our 'Laws of Physics' require time to be quantized. Humans, of course, had to quantize time before we could develop any theories based on the quantization of time. This is a straightforward 'chicken or egg' thing.

It seems to me that if Galileo had no preconceived idea of the measure of the passage of time (through the passage of seasons, day/night cycles, etc) he would not have been able to discover that the periodicity of a pendulum was constant. Depending on his state of mind, sometimes the swing would appear to take forever, sometimes it would appear to be shorter.

It also seems to me that Newton relied on a distinct concept of the passage of quantized time before hand, in order to develop many of his Laws. (The action/reaction thing, and gravitational attraction thing, would be valid, just not quantized as to the passage of time).

Building structural integrity and engineering calculations for stresses have no time component to them, if built by 'Rule of Thumb' (We built the last one this thick, and it is still standing. The other one was built thinner, and it collapsed.)

If this sentient species had no experience that the passage of time was measurable and periodic, would they ever develop a method to quantize it? Would they ever TRY to quantize it, if there was nothing in their environment that was consistently and repeatedly cyclic, that they needed to or even that they could consistently measure? What would that quantization look like?

I submit that this question is NOT the same as this, as many of the answers pertain to some form of naturally occurring cycle observable by humans, and /or are based on human circadian rhythms, which are not experienced by this species. This life form evolved without any ability to determine periodic time spans. Also, that question asks what would they use to quantize time, this asks what would their method of quantizing time look like? Would they even understand that time could be quantized into absolute divisions, that were equivalent between people and between events?

A further corollary question will be something along the lines of 'What would the Laws of Physics look like without time being quantized in the same constant periodic way that humans have quantized it?' But that is NOT in the scope of this question.

EDIT

The term 'quantized' comes from the digital field, not from quantum mechanics. The first use of quantum, in this regard, was in 1597. For a non-quantum-mechanics elaboration of the term, See Quantization (signal processing)

Quantization, in mathematics and digital signal processing, is the process of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite number of elements. Rounding and truncation are typical examples of quantization processes. Quantization is involved to some degree in nearly all digital signal processing, as the process of representing a signal in digital form ordinarily involves rounding.

EDIT

It is impossible for this species to develop any kind of calendar or any kind of measurement for a periodic recurring cycle. No days, months, years, seasons, hunger cycles, menstrual cycles, crop cycles, planting seasons, harvesting seasons, sleep cycles, or any other natural seasonal phenomena'. It takes three or more lifespans for the sky to repeat. The entire life of the planet has evolved without periodic cycles or rhythms. There is nothing to measure it with. There is no way to tell that this current crop is growing faster, slower, or just the same as the last crop. No way to judge or compare the pregnancy duration of the first child, with the length of the pregnancy for the last child. There is no metric, nothing to count, measure, or quantify. Any answer based on comparing the length of a current event to past or future events in any manner that does not end up with a number, or quantity, of some absolutely replicable unit of measurement, of the actual duration of the event, (so for instance two pregnancies, one then and one now, can be compared in length), is immediately discarded. Any comparison to any time duration in the past has to involve some kind of numerical count for the duration of each event.

Way back as far as 1597, the term 'quantum' was used to mean an amount of something. You could have a quantum of wheat, a quantum of distance, a quantum of liquid, a quantum of patience, a quantum of intelligence. a quantum of solace, a quantum of time. Some of these we consider to be measurable, some not. For instance, we talk about someone having more patience than someone else, or that one is 'losing patience', but what is the measure of patience that reduces to zero? For the things that we consider measurable, we have developed a system whereby they can be measured, counted, or quantified. But unless we have the CONCEPT that we can measure it, there has been no real consistent attempt to develop units of measure. The word 'quantum' is sometimes applied to the 'unit' of measurement - a quantum of distance is the foot, or the inch, or the meter, a quantum of weight is the gram, or the ounce, or the stone. Note that each one is different, even for measuring the same thing, so generically the term applied was 'quantum' meaning 'measurable amount'. But not all things that you could have a quantum of, could you have a 'measurable amount' of. Like patience. We can still talk about a quantum pf patience, without really any attempt at measuring, counting, or quantifying it.

Before we can measure, count, or quantify something, we first have to form the concept that it CAN be measured, counted, or quantified. My question is about developing this concept for time. (It could also be asked about temperature as well - we know something can be hotter or colder than something else, butt it took us a long time to develop the concept that the comparison could be MEASURED.) We can understand that time flows, that there are time passages, that the future is not the past, and we are in the present. Having a concept that time passes is not the same as understanding that it is measurable. Can we have the CONCEPT that time is measurable? We can not see, taste, feel, touch, or senese it except through memory, of knowing that the past happened. That is the process I call quantization and Miriam_Webster agrees with me.

Definition of quantize transitive verb

1 : to subdivide (something, such as energy) into small but measurable increments

Okay, so someone appropriated the term 'quantum' and proceeded to minimize it to absurdity, and it is now narrowly defined by some physics purists as ONLY meaning something to do with quantum mechanics. You do not have to agree with the definition or my terminology, you can come up with your own, but it what I am after is NOT the process of measuring, counting, or quantifying something, it is the process of realizing something CAN be measured or quantified in the first place. I do not want to know HOW it is measured, conted, or quantified, just that this species has the understanding and the concept that it CAN be measured, counted, or quantified.

We humans got the concept that time is something that could be measured, from the obvious day/night cycle. Historically, that is the first form of measuring time. We could COUNT days, and COMPARE the count between one event and another. We could RECORD the count, to compare durations in the present to durations in the past. It was not hard to conceive of the periodicity and the regularity of the day/night cycle. We could count the number of days from winter solace to the next winter solace, and see that the count was always the same. We liver f=through enough winter solaces to get the idea. It was natural for us to develop the concept that time could be measured, counted, and quantified.

But what if there was no calendar? Nothing to count? What natural phenomena would develop the concept that time was measurable in any meaningful fashion? ANswering this is fundamental to my corollary question - what would be the limitations on this species on their concepts in physics, based on their concept of the measurability of time? How would they quantify speed, for instance? They could easily compare two people in a race, and determine one was faster than the other, but what would their equations look like, without time? Distance, it is obvious to measure. But time?And a unit for time? Just as improbable, perhaps, for this species as a unit for patience is to ours.

So any answer HAS to include the reason why this species could even conceive that time could be measured in the first place. Again, any answer based on the premise that time HAD to be measured, in order to do something, is immediately disqualified. If there was no concept that time could be measured, there would be no concept that in any particular situation it was needed to be measured. You just don't do any activity that is time dependent if you do not understand the concept of measuring time. 'Meeting someone at such-and-such time' is just something this species would ever think about, because it just couldn't be done. ANything that puts a demand on this culture for time dependency as we know it is just anthropomorphizing Western culture on another alien species. No they DON'T have to behave the same way we do.

Sure, there are a lot of situations in which measuring patience is useful, but in all that, we still have no concept that patience can be measured in any absolute way, so we do not consider it a necessity to measure it absolutely. Would time be the same for this species? If not, what would trigger in them the concept that time could be measurable in a quantized (repeatable, replicable, consistent, understandable, quantifiable, countable, measurable, observable, any term you wish to use) way?

So okay, being specific about the question, and explaining it thoroughly and understandably takes a long time. But it has become obvious that TL:DR methods just don't work for many people.

EDIT

This did not make it into the previous edit.

Take, for instance, the development of the concept that length could be measured. I can measure a length of cloth with my arm, and mark it. Tomorrow, I can measure it, and find I get the same mark. I can measure ten arm's lengths, cut it off, and put it aside. I can take another piece of cloth, measure off ten lengths, and cut it. I can then compare the two lengths of cloth, and see they are both the same length. I can cut another piece of cloth at twenty arm's length, and notice it is as long as the other two pieces of ten arm's length combined. So I know length can be reliably, provably, replicably be measured. I can validate my measurement against something else. I can compare one measured piece with another that I made before and see that they are the same. I have no problem with the concept that length can be measurable. So absent any form of calendar or recurring periodic cycle, how do I get the same concept that time is measurable, can be quantized or sliced up or marked off so it can counted or quantified (whatever you want to call it) and assigned a numerical value, and this value will be the same for any other period of time of the same duration? Length I can see and touch and feel and store and compare between the past and the future. I can not do that with time.

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Obviously it would be trivial to measure time on this planet, just like anywhere else. The interesting aspect of the question is whether anyone would.

The existence of time is not the same as the existence of days or weekends or seasons. It's like how, on the open ocean or deep in the Sahara, distance exists, but "regions" do not (except in a very abstract way, which depends on an advanced science of cartography, which ultimately derives from our experience of natural boundaries like rivers, hills and coastlines).

A civilisation that arose on this planet would eventually understand time as a metric space – they wouldn't get far in science otherwise, and there's no particular reason why it would be difficult to figure out. But that doesn't mean the measurement of time would become fundamental to their understanding of the world the way calendars are for us. We don't experience color much differently just because we can measure the wavelength of light.

Medieval people didn't have a concept of meeting someone at "11am" – their "clock" consisted of sunrise, midday and sunset. If they didn't live in a village, they were probably hazy on days of the week. Things just happened when they happened. And that's with an extremely obvious natural "clock" that cycles every 24 hours and every 365¼ days, plus a secondary clock that cycles every 28 days.

On a planet with no natural clock, the idea of labeling time would be a very low priority indeed. It would be easy enough to understand the idea, but it wouldn't serve any practical purpose for farmers or sailors or ordinary people (unlike on Earth). If someone suggested "why don't we all synchronise our hourglasses and pay constant attention to them, and get stressed out if things don't happen at the right phase of the hourglass", the idea would seem bizarre and unpleasant.

In fact, it did seem bizarre and unpleasant to the working class when the Industrial Revolution first imposed the draconian plan of time that we now think of as normal. It took a good bit of brutality to get everyone on board. So perhaps that would end up happening when this tide-locked civilisation reached a certain point of development. Once society becomes a machine, there are technical advantages to being able to synchronise its remote parts.

That said, this world would certainly affect biology, and it may be that its people couldn't easily exist inside a calendar. If your sleep period naturally varies from 3 to 150 hours, that would create scheduling difficulties.

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TL;DR

If they develop a civilisation then yes, time will at some point become quantizied and that will be a major breakthrough in the civilisation. If they don't, then most probably not.

No civilisation

Let's face it. There are sentient animals other than humans on Earth. Currently the science claims most if not all vertebrates present some level of sentience. Yet, they have no idea of time.

Pets do not know how much time actually passed. Your dog will greet you as if you were away forever regardless if you have left for 10 minutes, for an hour or for a whole day. They may be aware of something being very long (e.g. someone is away for weeks or months where they normally are there most of the time or daily).

So if a non-civilised species on a heavily cyclic planet like that doesn't have any concept of time we can safely assume the safe to be true for any species anywhere in the Universe. Even if animals can synchronise it is due to instincts, not a concept of time.

Civilised species

From now on let me give your species a name of "nontemporians".

The situation is entirely different when we speak about developing a civilisation.

Assumptions

Just to make sure I have the right understanding of the question, let me first list some assumptions I take here.

  1. All laws of physics are the same, i.e. this is ours Universe, just nontemporians has no cycles natural to them to refer to. In other words all cycles of physics, including a cycle of planet going round its star, cycle of pendulum, half-lives of chemical substances, resonating cycles of crystals and so on all remain the same. The same is about the speeds, particularly speed of light and speed of sound are constant (even though a speed of sound in the local atmosphere may be different than that on Earth). Also all other laws apply so two object dropped from the same height to which atmospheric drag has dismiss-able effect will fall on the ground at the same moment.
  2. The nontemporians may not be interested in the amount of time but are aware and have the same perception of things being in the past, now or future. They can also relate to two things in the past and claim if they occurred simultaneously or not and if not, which one was earlier. In other words they are aware of something we would call a timeline of events. It might be though that they can't (at least initially) compare timelines of two different beings. On the other hand if they find a common point in the two timelines, they are aware that things before that point on both timelines happened before anything that happened after that point on any of the timelines.
  3. Nontemporians are social, i.e. they value having social interactions with other members of its own species. Note, I don't think a civilisation can develop without that. I assume there is a direct dependency between civilisation and sapience (i.e. you can't have one without the other). Of course I assume

Why

Sapience comes with curiosity. Species has to be curious in order to look for improvements rather than just accepting what the nature offers them. Sooner or later sapient species will try to explain things surrounding them and at some point hey will notice some dependencies. This way at some point of science development quantifiable time will become a theory that helps explain many things. As you have correctly noticed many discoveries require time in their understanding so discovery of time will be a real breakthrough.

How can I be certain? Let's look at thing that is as vague to us time to nontemporians: energy. We were exposed to energy since ever but we had no ways to measure it or quantify it. And actually it wasn't really needed. It was enough that you need to heat something more in order to change it from as solid to liquid. How much more - wasn't really important. Only with more and more theories it turned out that energy is everywhere and it sort of surrounds us, fills us and fuels us. Most of what we observe is related to energy. Movement changing, heating, chemical reaction, electricity, gravity - everywhere energy is involved. Milestones of civilisation were mostly related to energy. Using a fire was actually controlling of releasing chemical energy stored in coal (in various forms). Building a wheel was reducing energy required to move objects. When we (as a humanity) started understanding that, we were able to make most of the breakthroughs in the science.

At some point scientists started observing energy transmissions. They have noticed that the energy produced in one process can be used to fuel another process that requires energy. This led to things like engines but also to a definition off energy units - Joules and HPs.

We've been always using energy, eating, protecting ourselves from cold etc. Our civilisation is marked by the ways how we were learning how to control energy, even though we didn't even know that. Finally we started wondering what is there in common and how can we measure it and we found it.

How

At first time will not exist as a term at all. Yet quite early nontemporians will develop some forms of controlling moments in time. First they will notice som things - for example that they cannot tell the order of two things observed by different nontemporians and will start wondering if such order exists at all.

In the meantime nontemporians will start looking for synchronising their presence. They will notice that after some loud sounds they all gather at the same spot so they will check if calling loudly (or making loud noise in a different way) can help gather them into the same place. This way they will be able to socialise even more than when they were just scattered tribes. This discovery might lead to an improvement enabling to move from hunters-gatherers to farmers - there will be watchers set to observe crops grow. They will have some food piled and they will have to make loud noise then either the food runs out or the crops are ready to harvest.

This in turn will lead to having more and more individuals have time for thinking and observing (after all what a watcher will have to do? Eat, sleep and watch). One will notice, that when something loud is happening in the distance, they can always see it before they can hear it. Out of curiosity they will make an experiment to check if there is any relation and they will notice that if the objects are at the same distance, and you can see something at the same moment, you will also hear things at the same moment. This will lead to a discovery that when something is closer while the other thing is further and you can see something at the same moment then you can hear the nearer event sooner than the farther one.

There will be other observations - like stones falling at the same moment when dropped together.

Now there are few possible scenarios. Maybe they will even happen independently until someone, this Newton of nontemporians will find the relationship.

So one observation will be regarding to a pendulum or anything other cyclic (springy objects for example). Of course it will be interesting observation in itself that some objects repeat their position continuously. But then, just by accident someone will notice, that there are two pendulums, and one repeats its position exactly when the other different one repeats its position two times. Further tests will show, that you can construct pendulums that are repeating position every third third time or every fourth time of the "basic pendulum".

Then, again by an accident someone will notice that if they drop a stone at a specific position of pendulum and it falls to the ground after, lets say 5 repetitions of pendulum position repetition, then every next stone dropped from the same height will fall after the same amount of repetitions.

The seeing and hearing of the same distant event will follow. With more and more dependencies (time of event, height of dropping etc, etc).

And then some genius will understand that they have started measuring time and it all boils down to how much time have passed between two events. And that you can measure it by pendulum.

This discovery will open the ability to formulate things like speed, power etc., etc that eventually will allow building a time based civilisation. Just like we have energy based civilisation now, without really noticing it in our everyday life.

Time will have growing meaning. Giving some time for the land to recover after harvesting will enable more efficient usage of the soil. Knowing when to return nontemporians will be more efficient farmers. Having control on time passing they will also be able to synchronise even when not able to see or hear each other. It's actually difficult to explain all implications, but you can probably compare it to something like discovery of electricity in our civilisation.

There will be also other aspects. At some point nontemporians will start noting the position of stars. Some astronomer will then notice, that they repeat sometimes. Once time (and clock) is discovered, there will be a huge experiment run for generations, that will show that those star positions repeat in a cyclic way, starting an era of modern astronomy.

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Yes, human-like species would quantize time, because humans like to categorize and simplify everything, even when they don't have an accurate way to measure it. Wavelengths are on a continuous spectrum that humans had no way of measuring it for vast majority of their history, and they still came up with the concept of discrete colours. They didn't need accurate measurements for that.

The same thing would happen for time measurements: at first someone will notice that if you light a certain candle just as you put the turkey in the oven then the turkey will be ready at roughly the same time as the candle burns out (or something like that). They'll experiment more and notice that at least when it comes to chemical processes they'll match each other if nothing else (i.e. burning 1 gram of substance A takes twice as long as burning 1 gram of Substance B).

From there they can determine which things are syncronizable and which aren't, so even most of the world is not synchronizable at a certain point in development they'll know that some things are, and from there it's pretty trivial to standardize the measure of time (at least in regards to those things)

Same thing can be said about vehicles and speed: if you have devices that have constant-ish speeds and you know how to measure distance, you'll be able to conceptualize units of time as well.

So to reiterate, if they have civilization that is capable of chemistry or vehicle use, they will have concept of units of time, if only to compare things that aren't too variant (even if most of the world is).

Just to be extra clear:

Based on the comments the OP seems to question wether it would even occur to people on such world that time is something quantizable in the first place, and I'm saying that it doesn't have to. They don't have to think of time in any global sense to quantize it, if they match an hourglass to cooking a soft-boiled egg or pot-roast or whatever they can quantize "cooking units" and later "Alchemical Reaction Units" and much later they may discover everything else is connected to those units as well.

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