  We derive a dynamical bound on the propagation of correlations in local
random quantum circuits - lattice spin systems where piecewise quantum
operations - in space and time - occur with classical probabilities.
Correlations are quantified by the Frobenius norm of the commutator of two
positive operators acting on space-like separated local Hilbert spaces. For
times $t=O(L)$ correlations spread to distances $\mathcal{D}=t$ growing, at
best, diffusively for any distance within that radius with extensively
suppressed distance dependent corrections whereas for $t=o(L^2)$ all parts of
the system get almost equally correlated with exponentially suppressed distance
dependent corrections and approach the maximum amount of correlations that may
be established asymptotically.
