What s the current thinking on being able to run Schrödinger's cat backwards? How can time be reversed on something in an indeterminate state?
The Copenhagen (traditional) interpenetration of quantum mechanics posits that when a system gets large enough [0] these superimposed states collapse down to a single state. This interpenetration does pose a problem with time reversibility, as if you run time backwards, then forwards again, it is not guaranteed that you will arrive in the same state after collapse.
A more modern interpenetration is Many Worlds. In this interpenetration, there is no collapse. So when we observe the cat, the universe is in a superposition of us having observed a dead cat, and us having observed a living cat. When we run time backwards, we end up at the point where we put the cat into the box, as you would expect. If we run time forwards again, we arrive at the same superposition of observing a living cat, and observing a dead cat.
Without looking at the math, the ability to go from a superimposed state to a pure state may seem like it presents an asymmetry, but the equations allow (and we have experimentally done) this in forward time already. As it turns out, the mechanism that allows for this to happen (interference) is the essence of quantum computation.
[0] The standard phrasing is is "gets measured". I read this as "gets large enough" because we could, in principle, consider the measurement device to be part of the system.
Certain observations are probabilistic, and the correlations in them predicted by quantum mechanics cannot be reproduced by ascribing them to the traits of particles.
It means that at the lowest level, the universe is (barring faster-than-light communication, generally regarded to be false) "truly random". (No concept of an "observer" needs to be devised in order to arrive at this conclusion).
A many-worlds interpretation (which is consistent with QM) is basically the same as saying the result is random, because you can't predict exactly which branch any particular interaction will follow.
If you want to talk about "truly random", please say more.
Some theories rule out what you may be calling "truly random".
How would the reversing process know the direction and momentum to reverse for each particle?