  We study the relations between the recently proposed machine-independent
quantum complexity of P. Gacs~\cite{Gacs} and the entropy of classical and
quantum systems. On one hand, by restricting Gacs complexity to ergodic
classical dynamical systems, we retrieve the equality between the Kolmogorov
complexity rate and the Shannon entropy rate derived by A.A.
Brudno~\cite{Brudno}. On the other hand, using the quantum Shannon-Mc Millan
theorem~\cite{BSM}, we show that such an equality holds densely in the case of
ergodic quantum spin chains.
