A few obvious answers are at hand: The definition of Time shows why Quantum is more fundamental than Time. Trivial, but deep. As observed!
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Time is the ticking of a light clock, a device made of two separated mirrors, between which light rebounds, providing a periodic transfer of momentum, which can be measured. That makes time a local concept. The notion dates back to Lorentz, who used it implicitly, and Henri Poincaré, who defined it explicitly, building “Relativity” around it (19C)… Many years before a celebrated parrot…
More subtle, and so far unnoticed: Defining time from light, as we have to, there is no other way, makes time less fundamental than fundamental Quantum processes, as light is the fruit thereof, and not reciprocally. This is philosophically consequent as far as the speed of light being an absolute notion. It also means that time is less fundamental than space. Indeed, time is used as one parameter group of transformation in fundamental Quantum mechanics.

The speed of light is a local, loop concept. The speed of Quantum collapse or entanglement is a global concept… which can’t be limited in any sense by a local, point-like notion (such as the speed of light).
Both Relativity and Complex Numbers pop out of the physics of light, which then constitute their physical embodiment.
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The preceding are broad, contextual, extremely powerful (“philosophical”) arguments. They give good reasons to believe that Quantum Mechanics is more fundamental than time, and thus, would violate time.
However, the devil, physics is in the details. Can we check these broad ideas? Is there a violation of time somewhere? Yes!
CPT symmetry changing charge C, parity P, and Time T simultaneously leaves physics unchanged: it is a theorem of Quantum Filed Theory. However CP violation was observed in 1964 (Nobel 1980). That means time violation had also to occur, to compensate.
The preceding is the sort of very general reasoning physics is always driven by. A related example is an argument about antimatter. There is matter around, no antimatter (or, more exactly, ephemeral). According to the Big Bang, the LCDM model, there should be as much of one as of the other. If there is not, as we observe, the only explanation is CP violation. Thus the LCDM model, in combination with very little antimatter shows that there is time violation, time is not a fundamental symmetry. So much for time travel.
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Notice I did not mention entropy. The second law of thermodynamics says it augments… with the number of “states”, as in thermodynamical states. However, it’s hard to define “states” … In Quantum Physics, there are “states”, of course, even “eigenstates. However those states are relative to a particular situation, the experiment at habd Niels Bohr was (rightly) obsessed about. Now one can make analogies with the surface of black holes, speak about “information”, etc… All interesting, but far-fetched hypotheses by philosophical, or mathematical analogies. I do not want to deride those, as I myself make some… of my own… I claim mine are better, as they rest on clearer and simpler, deeper principles (for example no interaction at a distance at infinite speed).
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Where does all this want to go? If Quantum is deeper than light, and Quantum itself is enacted by Quantum Entanglement, we get to the conclusion that QE is the real architecture of the universe: both Quantum and Relativity are macro, emerging consequences… Hence the necessity for a Sub Quantum Physicsal Reality, SQPR.
We think, therefore we hope.
Patrice Ayme
P/S: Notice in passing that “quantum” was elevated from my grammar to the level of time…