Societies advance when they become more internally cohesive, trusting, egalitarian and merit driven. They fall when they lose those traits.
An old blog post of mine expands upon this
Military success, increased trade, growth of science, technology and arts are all epiphenomena of the social and political changes.
Rome is a canonical example. Rome became a superpower because it was a representative republic with leadership and advancement based more on merit than lineage. Lineage based societies select leaders of all kinds based largely on lineage. You get to be king because your father was. If the leaders are idiots, your stuck with them until they die. Merit driven republics can choose leaders based on performance and remove them in the short term if the fail. Conversely, they can reinstated if proven correct e.g. Fabius vs Hannibal.
Rome also allowed a much higher percentage of it's population to bear arms meaning that it could field massive armies compared to surround populations. This gave Rome more military manpower per unit of population that cultures that allowed only a fraction of their population to be armed. Rome became much more militarily "dense" than its competing polities.
When the Republic fell, so to did merit promotion and high numbers of military service. The Empire continued by inertia but it was already doomed.
It's dispositive that that the Republic expanded massively compared to starting size, growing several times over from a mere county sized area to cover half the Mediterranean world in a little over two hundred years. The Empire expanded only a about a third and reached its maximal extent in the first century.
It's easy to see why. Whereas the Republic trusted it's soldiers as sought to see them well armed, trained and led, the Emperors feared them (rightly so) and kept them well paid, but poorly armed trained and led. The imperial armies grew more expensive and political while growing less and less effective. Worse, the percentage of the population who could be armed or serve plummeted. Rome relied more and more on hired auxiliaries who were usually "barbarians" with little loyalty to Rome.
From the time of Marius onward, the rule of law in Rome progressively failed with economic success relying more and more on success and connections in politics which in turn grew more and more violent. Romans spent progressively more time fighting each other than anyone else. Merit promotion died.
By the end of the second century Rome had inherited vast borders but had lost the social and political systems that fielded the armies to defend them. Trade faltered as property, contracts and profit became matters of political whim. With the fall of trade came less taxes and less military spending. Area after area become isolated and indifferent to affairs in the general empire.
In last century, "Roman" armies and their opponents where largely indistinguishable. Rome didn't fall to barbarians as much as become them. Rome had no technological, social, economic or political advantage over the surrounding societies and they just oozed over the borders.
In the end, Rome was just another rigid, hierarchal, lineage based society among many. It simply dissolved into the noise.
The pattern is repeated time and time again. Athens, Rome, Carthage, Genghis Khan, Venice, Florence, Lisbon, Swiss Cantons, Dutch Republic, England, America.
In each case, the societies became more merit driven with decentralized political power and surge in trade, science, art and military success. The success of some is just staggering. Venice and the Dutch Republic particularly stand out. Both were small societies built on river deltas who literally didn't even have land. Yet, they ended up dominating their regions and influencing much of the world.
But such societies do not seem stable. Their dynamic periods last only 100-200 years followed by decline into mediocrity. Meritocracy is counter to our genetic programming to favor our kin over others. In the end, riches, arrogance and complacency lead us to let our guard down and we revert to the lineage pattern.
I would note that societies generally never recover from decline. Instead the torch of progress passes to another. The Renaissance wasn't the comeback of Rome, it was isolated to a shockingly small number of people in a few Italian cities for under a century and then it promptly shifted to Northern Costal Europe. The progression went Venice --> Florence --> Lisbon/Geneva --> Dutch Republic --> England --> America --> ?