  In this paper we consider the nonlinear inverse problem of phase retrieval in
the context of dynamical sampling. Where phase retrieval deals with the
recovery of signals & images from phaseless measurements, dynamical sampling
was introduced by Aldroubi et al in 2015 as a tool to recover diffusion fields
from spatiotemporal samples. Considering finite-dimensional signals evolving in
time under the action of a known matrix, our aim is to recover the signal up to
global phase in a stable way from the absolute value of certain space-time
measurements. First, we state necessary conditions for the dynamical system of
sampling vectors to make the recovery of the unknown signal possible. The
conditions deal with the spectrum of the given matrix and the initial sampling
vector. Then, assuming that we have access to a specific set of further
measurements related to aligned sampling vectors, we provide a feasible
procedure to recover almost every signal up to global phase using polarization
techniques. Moreover, we show that by adding extra conditions like full spark,
the recovery of all signals is possible without exceptions.
