(work with S Pappalardi)
In the past few years there has been considerable activity around a set of quantum bounds on transport coefficients (viscosity, conductivity) and on chaos (Lyapunov exponents), relevant at low temperatures. The interest comes from the fact that AdS/CFT Black-Hole models seem to saturate all of them. I shall discuss the simple case of bosonic systems whose lowest energy is a degenerate manifold, and in particular free motion on a curved space, the Hamiltonian being just the Laplace-Beltrami operator. Examples are quantum hard-sphere liquids and quantum spin liquids. In this context the bounds are approached and are consequences of the uncertainty principle, and one understands the mechanisms whereby quantum mechanics enforces them. For a system to saturate the bound, it appears as a necessary condition that at each temperature there are some degrees of freedom that are still classical, and some are on the verge of being affected by quantum effects.