The wavefunction obtained by the Schrödinger equation is indeed time dependent, so it is possible for the probability of measuring a certain eigenvalue of a physical quantity described by an operator on the wavefunction to change over time. I don’t quite get what you are trying to say here. Are you talking about whether quantum entanglement breaks relativity? Measuring for example one electron spin up will get you the information that the other one (maybe a few light-years away) is of a certain other state (maybe spin down). This does not allow to transfer a message between the two locations, as measuring the state on the senders end is still completely random. They would have to send a lookup table which is still only possible with the speed of light