  In this paper we translate the two higher levels of the Ergodic Hierarchy
[1], the Kolmogorov level and the Bernoulli level, to quantum language.
Moreover, this paper can be considered as the second part of [2]. As in paper
[2], we consider the formalism where the states are positive functionals on the
algebra of observables and we use the properties of the Wigner transform [3].
We illustrate the physical relevance of the Quantum Ergodic Hierarchy with two
emblematic examples of the literature: the Casati-Prosen model [4], [5] and the
kicked rotator [6], [7], [8].
