  We prove bounds on the minimal time for quantum messaging,
propagation/creation of correlations, and control of states for general lattice
quantum many-body systems. The proofs are based on a maximal velocity bound,
which states that the many-body evolution stays, up to small leaking
probability tails, within a light cone of the support of the initial
conditions. This estimate is used to prove the light-cone approximation of
dynamics and Lieb-Robinson-type bound, which in turn yield the results above.
Our conditions cover long-range interactions. The main results of this paper as
well as some key steps of the proofs were first presented in [36].
