  We study the spreading of quantum correlations and information in a
one-dimensional quantum spin chain with critical disorder as encoded in an
infinite randomness fixed point. Specifically, we focus on the dynamics after a
quantum quench of the R\'enyi entropies, of the mutual information and of the
entanglement negativity in the prototypical XXZ spin chain with random bonds
and anisotropy parameters. We provide analytic predictions in the scaling
regime based on real-space renormalization group methods. We support these
findings through numerical simulations in the non-interacting limit, where we
can access the scaling regime.
