I'll answer this just to get a feedback about my own understanding of this (probably much more complicated than I think) subject.
The wave function will always evolve unitarily according to Hamiltonian. If the state of initial preparation (or a state after collapse) happens to be an eigenstate of the subsequent measurement - you will measure a per-determined eigenvalue. In other words (the story my intuition invented to settle this stuff inside my head), if you prepare (or measure) the system in a state which does not contain any undetermined information for subsequent measurement - you can predict the result of this measurement.
Once the measurement is done the wave function collapses. What does it mean? A lot of bla-bla, metaphysics, religious and cultural discussions and etc. I did not really understand this collapse completely. However, I know that this collapse brings a wave function to an eigenstate of the measured observable. This provides the following information about subsequent measurement:
- If the eigenstates of subsequent measurement are identical to the eigenstates of the previous measurement (I suspect that the right formulation of this is that there are "one-to-one" and "onto" mapping between these sets of eigenstates) - see the first paragraph
- If the eigenstates sets are not exactly identical, but there is partial correlation - you can predict some probabilities of the subsequent measurement
- If the eigenstates sets are "independent" - you get no information about subsequent measurement's result
In other words (my intuition is such a story-teller!), the more correlation there is between this observable and the subsequent one, the more information you can get about the subsequent measurement.
All the above feels reasonable as long as Hamiltonian does not change. If there are external factors which change Hamiltonian (as I believe the case in real measurements), there are no guarantees whatsoever. However, and this is a pure speculation, I guess that if one can predict the evolution of Hamiltonian in time - some predictions about subsequent measurements may still be made (unless the observables are completely independent).