  In the context of the Many-Body-Localization phenomenology we consider
arbitrarily large one-dimensional spin systems. The XXZ model with disorder is
a prototypical example. Without assuming the existence of exponentially
localized integrals of motion (LIOMs), but assuming instead a logarithmic
lightcone we rigorously evaluate the dynamical generation of $ \alpha$-R\'enyi
entropies, $ 0< \alpha<1 $ close to one, obtaining a $\log \, t$-law. Assuming
the existence of LIOMs we prove that the Lieb-Robinson (L-R) bound of the
system's dynamics has a logarithmic lightcone and show that the dynamical
generation of the von Neumann entropy, from a generic initial product state,
has for large times a $ \log \, t$-shape. L-R bounds, that quantify the
dynamical spreading of local operators, may be easier to measure in experiments
in comparison to global quantities such as entanglement.
