@DisasterlyDisco raised some very good points regarding the dimensionality of space-time and @AdrianColomitchi also points out fundamental issues with the presentation. With regards to this I'd like to elaborate while providing some insight into how to properly use the terminology of quantum mechanics.
In physics, words like "state", "dimension" and "quantum" have very specific mathematical meanings. Thus a system based upon these concepts used incorrectly, to the trained ear, will not sound coherent and hence not realistic.
In quantum mechanics, a "state" refers to a mathematical formation known as a wave-function. The wave function contains information about the probability of that system having particular quantum number. These quantum numbers correspond to things called observable, which we can measure.
Now this is all very abstract (quantum mechanics is notorious for this), and so here's an example which hopefully will clear up how to use the terminology.
A basic example: demonstrating how to apply quantum principles
Imagine you just have one particle: If this were classical mechanics its easy, you can describe everything about this physics of the particle with a few quantities: where it is at :(x,y,z,t), its momentum ($p_x$,$p_y$,$p_z$) and its total energy: H = T+V, where T is the kinetic energy you can get from its mass and momenta, and V is the potentials that the particle is under the influence of. While this seems complicated, once you know all these quantities, you know everything there is about the particle, furthermore once you know these things, you can predict exactly how the particle will move: its velocity and its acceleration.
This is not true of quantum particles, which means that they must be described with wave-function. Lets look at a really simple example:
Consider just one quantum particle which is not under the influence of any potential. We would describe it with a wave function, to really simplify it lets restrict its motion to one dimension (the x axis):
So what is this particle doing? The answer, we don't know, but we can make an educated guess. For starters the particle might be propagating (or moving) in either the +x direction or the -x direction, so we say that its total wave function is the super position of these two "states":
$|\Psi \rangle = |\phi\rangle_+ + |\phi\rangle_-$
Now, this does not have the meaning (which is unfortunately the most pop-sci way to talk about this) that the particle is moving both in the +x and -x directions, what this means is that the particle has probabilities of moving in each of the +x and -x directions, and we can't know for sure until we measure the particle.
So we have the wave function, but where is the particle at. I don't have a x coordinate yet. How do I get it? The answer is that its buried in the wave function. So to get it out of the wave function you apply the position operator and perform what is called the "inner product" over an interval [$x_a$,$x_b$]:
$\langle \Psi|X\Psi \rangle$
Now I know the probability that the particle is between $x_a$ and $x_b$. But I still don't know where it is at. That's because that as good as we can get with quantum mechanics.
But what about its momentum, the answer same thing:
$\langle \Psi|P\Psi \rangle$
Don't worry about the math of how to do these calculations. That would require taking a university level course in quantum mechanics, and fortunately is not necessary to understand how this work.
So if we can't know the energy here, how can we ever talk about energy in quantum mechanics if we can only get probabilities?
The answer to that is the mystery of quantum numbers: in both quantum and classical systems particles can be anywhere. But in quantum systems, unlike classical systems, they cannot have just any 'ol energy value but must contain specific units of energy which may increment/decrement in discrete levels.
Again however, we can't know for certain the particles exact value, but only until we "measure" it. Measuring doesn't have to involve a scientific instrument, it just refers to an interaction the particle undergoes which collapses the wave function, and the "actual" values can be observed.
Now if you were keen on the math you might have noticed something a little strange here; I stated that taking the inner product of the wave function, when an observable operator
gets applied, returns a probability for the observable. But in our example above, the inner product of the wave function would look like this:
$\langle \Psi | \Psi \rangle$
which would expand to this:
$(\langle\phi|_+ + \langle\phi|_- ) (|\phi\rangle_+ + |\phi\rangle_-\rangle)$
and expanding completely:
$ _+ \langle\phi|\phi\rangle_+ \ + \ _+ \langle\phi|\phi\rangle_- \ + \ _- \langle\phi||\phi\rangle_+ \ + \ _- \langle\phi|\phi\rangle_-$
Our math only works if, under the implication that the states are orthogonal $\ _+ \langle\phi|\phi\rangle_- \ = \ _- \langle\phi|\phi\rangle_+ = 0$
But +x and -x are clearly not orthogonal, what gives.
This is because the states of the particle moving in these directions are orthogonal not the actual directions themselves.
How this applies to the question
How does this answer your question? Lets go piece-by-piece:
I have put together three dimensions of time, and use quantum time states so time does not blur together
Using the example as reference, quantum states refer to wave-functions and not to physical dimensions such as x,y,z. While wave-functions incorporate physical dimensionality into their construction to be sure, they they only apply to the quantum particles themselves. As shown in the example, qunatum mechanics does not imply that individual states blur together until observed, but serve to provide probabilities regarding the actual values of the observables such as position and momenta.
We would not talk about a quantum "time state", just like there is no quantum position states or quantum energy state. There is a quantum state, which may be a super-position of individual mathematically "orthogonal" quantum states.
The entire universe is treated like a time object, so the fact it gets wider or thicker doesn't violate conservation
There about 16 conservation laws of which 6 are always true. If a conservation law is broken, we say that there is conditions for the law to be violated and we refer to this as symmetry breaking. So what is a "time object"? And which conservation laws does being a "time object" prevent the universe from breaking?
My understanding says this should work and not violate generally-understood current theoretical physics [regarding the three dimensions of time]
The usage of the word dimension here, is not consistent with the mathematical definition. The mathematical definition of dimension reference to the most basic element required to define a single point within a mathematical space. Your three "dimensions" are really three restrictions on how time works in your story, lets examine each one of them:
Causality: The first dimension of time is causality. Cause follows effect in a classical "timeline." You can't alter your own timeline. Each moment of time represents a quantum time state and a separate strand of time.
I agree that to form a logical system, you cannot violate causality. Not allowing your characters to alter their own timeline would be a good way to prevent causal paradoxes. However, the last sentence, doesn't really make sense as regards the usage of the physics language. See above regarding the meaning of "quantum state". To illustrate, a moment of time would be a meaningless formation for a quantum state, equivalent to saying "the quantum states of x = 3".
Synchronicity: The second dimension is synchronicity. All points are offset by a moment of time, so all points of time are happening simultaneously in separate quantum time strings. You can "travel" to your childhood and change things, but the changes only affect that time state, so when you go home your universe is still the same.
A "moment of time" as measure by whom? All time is relative to specific frames of reference in our own universe. This does not mean that time does not obey certain rules, it certainly does, but these rules allow for different frames of reference to measure time intervals differently than one another.
What is a "seperate quantum time string"? Is this a fancy way of referring to one universe within the multiverse? If so, how are these multi-verses connected. Look-up Brian Green's options for how multiverses can exist if you are interested in how to construct a multi-verse which doesn't violate the known laws of physics.
Probability: This is my least worked-out dimension and the least critical for plot. All observed possible quantum events happen, and the more they are observed, the thicker the observed universe gets.
What is the observed universe? This is a way of saying multiverse, or talking about the literal observed universe, which IRL consists of all the stars and galaxies we can see from earth. Does this mean that the actual universe somehow gets new universes appended onto it? From the previous descriptions it seems that you want multiverse.
Is the meaning that all possible quantum events occur, just not in the same universe, but spread through out the multiverse? If so, then I'd consider the implications of this. For instance, if a probability of a quantum number, as determined by a quantum state consisting of a basis of two states, is 0.3 for state A, versus 0.7 for state B, does this mean that in one universe state A is the actual state and for another that state B is the actual state (two overall). Or, does this mean there are seven universes with state B, and only three for which state A is the actual.
Also remember, that quantum states are very distinct from human decisions (for which the physics is currently unknown). Common trophes might regard monumental and pivotal moments in history: for instance, a president ordering a nuclear strike. In one universe, the strike is called off at the last moment and the world of tomorrow appears a few years later, in another universe the planet is in the depth of a nuclear winter with human life on the brink of extinction. While it sounds good, even if the multiverse hypothesis is correct, quantum physics does not support these kind of outcomes resulting from the collapse of quantum states.
Another similar concept is found in "random" events being cast as quantum events. So for instance, the villain has the hero on his knees and is ready to vaporize him, but decides to be "merciful" and toss a coin to determine his fate. The coin lands heads and the hero lives to fight another day, but the evil mastermind cackles...knowing the hero's been vaporized in another universe...Except this isn't correct at all. Tossing a coin is a classical event, which only appears random from normal level of baysian inference and is not decided by collapsing quantum states. To fully explain this would require an answer all of it's own.
The expanding universe displaces existing timespace, so nothing is created, only transformed from unfixed pluripotent reality (chaos) to fixed reality.
How is time-space transformed, allowing multiple actualized states to exist simultaneously with the creation of independent realities? And how does this not violate the first point regarding causality? What is the mechanism by which the "unfixed pluripotent reality" becomes "fixed reality"? Does this occur for every quantum event? And if so, how are the other "time strings" kept separate?
I hope this was just enough quantum mechanics to help you to understand the terminology and demonstrate how the principles are applied without being bogged down with too much math.