I had the same thought a few years before you did. As I understand it - and I'm still thinking about this one - your question is excellent - when this happens out and about, all information about what the system was is lost within the system itself when it 'collapses' due to a measurement. There is nothing in the system itself which retains information about its history. Hence the 'random' choice of one of the eigenstates and values.
Suppose there is the case of a system which just absorbs a photon - this will change the wavefunction by shifting its energy up a photon's worth, but it is now in a new eigenfunction of energy. This also therefore changes its momentum and thus its position is differently unknown. But how do we know of these unknowns? We can only find out by the measurement of the system - so how do we know about its new energy eigenstate? We'd have to make another measurement. Suppose we did so by chucking another photon in its direction and conveniently it chucks one back out in our direction which we absorb (and we assume nothing else is interacting with the system). Now our energy eigenfunction has shifted - perhaps we can tell this by being the measuring apparatus (say we are a frog with a good eye for one photon).
So the two systems have interacted through that single-photon interaction. And because the frog's eye saw a blue photon, we know it was that energy that came out of the system rather than the energy that a red photon would indicate. So we know the system just dropped from whatever state it had evolved to (a mess of its energy eigenstates) at its previous energy eigenlevel down one blue photon eigenlevel. If we knew already (from a previous measurement) what that eigenlevel was, we know what it just dropped to, and also by The Laws of Quantum Physics, that it's once again supposedly now time-evolved to another mess of its eigenstates. But we can't know that something else didn't do something else with it in-between, like, say, the exact same thing we were doing to it but from a different direction, and perhaps in the absorption and emission of a red photon instead - and you see the difficulties about knowing what's going on.
And that is the simplest version of how all such interactions must take place. But we didn't get any information about what the system had evolved to before that blue photon came to us, and none is retained in the system. Some would say that information never really existed at all (because it's only information upon interaction), and therefore the system was never in any state other than an eigenstate by reasoning of what information can be gained about a system.
I would then ask about the effectiveness of quantum computation - and then be told it's multiple universes, in each eigenlevel-numerated of which the system is in the appropriate eigenstate, but this still begs the question of how the universes know how to communicate such that they don't accidentally send the information about one 'collapse' to the wrong universe, or if you like, what controls the propagation of universes, and how it is that we can get the result we're looking for out in this universe by dragging back the interference from all the others. Actually, the loss of information in this universe is clearer to see in this model, because the other eigenstates are so well-removed in the other eigenlevel universes (as well as in their other times, if that's still bothering you).
In the case of the laboratory described in a previous example, we know what is going on there because we have set up that experiment and know its properties. We can therefore 'reverse' what happened to it because what we are doing in that case, is exactly the same as starting the experiment again. Note it isn't the same experiment - there is no such thing in quantum physics - it's done at a different time.
I have perhaps been a little verbose to detail my description. But I think such discussion require such attention. I reiterate that your question is excellent, and that I, too, wonder about this time directionality problem and whether it is all controlled at precisely this level. Controlled by what is, I think, the next big step in physics.