Does gravity affect the time evolution of a QM wave function? We know that the Schrödinger equation describes the time evolution of a wave function, but how does gravity affect that evolution? For example, does the wave spread slower in a strong gravitational field that in a weaker one since a clock of the system runs slower?
 A: There is no theory of quantum gravity yet, but we can say that also in quantum mechanics, gravitational time dilation is affecting mass particle quantum systems. This fact is already used in quantum physics: The measured time (of the laboratory clock) is the time after gravitational time dilation (redshifted with respect to proper time), and from this measured time may be retrieved the proper time of the quantum system if we know the gravity forces which are acting on it.
A: If you're talking about the non relativistic Schrodinger equation
$$
i \hbar \frac{d}{dt} \psi = - \frac{\hbar^2}{2m} \nabla^2 \psi + V(x) \psi
$$
then a gravitaional field effects the particle by changing the external potential $V(x)$ to be the gravitational potential energy at position $x$. In this case, no, there is no time dilation present.
If you want a relativistic theory of quantum mechanics, then you have to use quantum field theory. You could couple a quantum field to a fixed background metric $g_{\mu \nu}$, and in that case yes, there would actually be gravitational time dilation. The "wave function" of a particle is not perfectly well defined in curved space time, but I would say yes, for most intents you could say that the "wave function would spread slower" higher in a gravitational field.
A: In the context of GR gravity is space-time curvature.  Thus one can discuss the effect by developing and solving the QM equations one a curved manifold, using the metric tensor for the Schwartzchild metric.  There is some theoretical research connected to this.  One area of research in the 80's involved this approach in the context of demonstrating that trapping particles in curves and curved surfaces generated a geometry based potential well, leading to trapping particles in the highly curved regions.  I realize this is not what you are asking about, but these articles are applicable to real solid-state systems and the approach is noteworthy as it serves as a toy model for figuring out what you want to do.  Another area of research, similar time period, is developing QFT on curved space-time manifolds.  There are actually some text books available on this topic that might interest you.  I cannot recall the author, but google the phrase "Quantum Field Theory on curved space-times" and you'll find them.
