  The common idea behind complexity reduction in physical systems is separation
of information into "physically meaningful" and "safely ignorable". Here we
consider a generic notion of such separation -- implemented by coarse-graining
the state-space -- and address the question of what information is indeed
safely ignorable if we want to reduce the complexity of dynamics. The general
condition for reducibility of dynamics under coarse-graining will be presented
for stochastic and quantum systems. In the process we develop the quantum
notion of state-space coarse-graining that allows to marginalize selected
degrees of freedom. One of our main findings is that there is a broader class
of symmetries, beyond those that are considered by Noether's Theorem, that can
play a role in the reduction of dynamics. Some examples of quantum
coarse-grainings and the reduction of dynamics with symmetries will be
discussed.
