Chaos and Complexity of quantum motion

The problem of characterizing complexity of quantum dynamics in particular of locally interacting chains of quantum particles will be reviewed and discussed from several different perspectives: (i) stability of motion against external perturbations and decoherence, (ii) efficiency of quantum simulation in terms of classical computation and entanglement production in operator spaces, (iii) quantum transport, relaxation to equilibrium and quantum mixing, and (iv) computation of quantum dynamical entropies. Discussions of all these criteria will be confronted with the established criteria of integrability or quantum chaos, and sometimes quite surprising conclusions are found. Some conjectures and interesting open problems in ergodic theory of the quantum many problem are suggested. PACS numbers: 05.45.Pq, 05.45.Jn, 03.67.Mn, 05.60.Gg Invited review article for special issue on Quantum Information