  Using tools from classical signal processing, we show how to determine the
dimensionality of a quantum system as well as the effective size of the
environment's memory from observable dynamics in a model-independent way. We
discuss the dependence on the number of conserved quantities, the relation to
ergodicity and prove a converse showing that a Hilbert space of dimension D+2
is sufficient to describe every bounded sequence of measurements originating
from any D-dimensional linear equations of motion. This is in sharp contrast to
classical stochastic processes which are subject to more severe restrictions: a
simple spectral analysis shows that the gap between the required dimensionality
of a quantum and a classical description of an observed evolution can be
arbitrary large.
