  Stochastic finite-state generators are compressed descriptions of infinite
time series. Alternatively, compressed descriptions are given by quantum
finite- state generators [K. Wiesner and J. P. Crutchfield, Physica D 237, 1173
(2008)]. These are based on repeated von Neumann measurements on a quantum
dynamical system. Here we generalise the quantum finite-state generators by
replacing the von Neumann pro jections by stochastic quantum operations. In
this way we assure that any time series with a stochastic compressed
description has a compressed quantum description. Moreover, we establish a link
between our stochastic generators and the sequential readout of many-body
states with translationally-invariant matrix product state representations. As
an example, we consider the non-adaptive read-out of 1D cluster states. This is
shown to be equivalent to a Hidden Quantum Model with two internal states,
providing insight on the inherent complexity of the process. Finally, it is
proven by example that the quantum description can have a higher degree of
compression than the classical stochastic one.
