It is the same for quantum mechanics, in technical terms the time evolution of a quantum system is unitary. It is the uncertainty principle that prevents you from learning the exact state of a system and therefore perfectly predicting its evolution into the future or past. The underlying mathematical structure nonetheless allows you to do exactly this assuming you exactly knew the state of the system at one point in time.

There is of course the infamous collapse of the wave function that seems to turn a quantum state into a classical state during a measurement and this with different probabilities for different outcomes. This collapse of the wave function is not unitary and there are no good reasons to assume that a measurement is in any way special and should violate the laws of quantum mechanics which have otherwise been shown to work with incredible precision.

Therefore the collapse of the wave function is probably not a thing but just an apparent artifact of our ignorance of the details of the measurement apparatus which just is a gigantic quantum mechanical systems with billions and billions of particles.

Another way to think about that are symmetries and conserved quantities. The physical laws are believed to have certain symmetries, for example time and space translation symmetry or rotation symmetry, i.e. a physical experiment should yield the exact same result no matter whether you do it here or five meters to the left, today or in three weeks, with the laser beam pointing north or east.

Space translation symmetry, for example, means that if you change all you initial say x coordinates by the same amount the calculations should nonetheless yield exactly the same result as a calculation with the unchanged coordinates besides that the final state will of course have the same offset in its x coordinates.

If you demand that your laws of physics obey such symmetries and do some math you will find that each such symmetry inevitably gives you a quantity that does not change over time, i.e. that is conserved. That is Noether's theorem and extremely general, i.e. (almost) no matter what laws you imagine, a symmetry will always give you a conserved quantity.

The conserved quantity for space translation symmetry is what we call momentum, for rotation in space it is angular momentum and for time translation it is energy. But now that we have conserved quantities it is now impossible that two states differing in any of the conserved quantities will ever evolve into the same state or have ever been in the same state because that would require the change of a conserved quantity.

So to summarize, very basic assumptions like isotropy and homogeneity of space and time inevitably lead to conserved quantities which in turn prevent certain state evolutions that would violate those conservation laws. I am not sure if those constraints force time reversibility onto you but I assume they do.

Another commenter mentioned T-symmetry, that is that the physical laws remain the same if you reverse the time axis, i.e. multiply the time coordinates by minus one flipping past and future. This is of course not the same as time translation symmetry which is about adding to the time coordinates and it is also different from the reversibility I mentioned before, i.e. the ability to retrodict past states which does not require that the evolution into the past is symmetric with the evolution into the future. T-symmetry is believed to be violated for very good reasons but that has not yet been shown experimentally although it follows from the observed CP-violation and the assumed CPT-symmetry.

And I am not a physicist, so while I believe everything I said is accurate or marked as me not being sure or not knowing it may still contain some - hopefully minor - errors.