  We employ $(1 + 1)$-dimensional quantum cellular automata to study the
evolution of entanglement and coherence near criticality in quantum systems
that display non-equilibrium steady-state phase transitions. This construction
permits direct access to the entire space-time structure of the underlying
non-equilibrium dynamics. It contains the full ensemble of classical
trajectories and also allows for the analysis of unconventional correlations,
such as entanglement in the time direction between the "present" and the
"past". Close to criticality, the dynamics of these correlations - which we
quantify through the second-order Renyi entropy - displays power-law behavior
on its approach to stationarity. Our analysis is based on quantum
generalizations of classical non-equilibrium systems: the Domany-Kinzel
cellular automaton and the Bagnoli-Boccara-Rechtman model, for which we provide
estimates for the critical exponents related to the classical and quantum
components of the entropy. Our study shows that $(1 + 1)$-dimensional quantum
cellular automata permit an intriguing perspective on the nature of classical
and quantum correlations in out-of-equilibrium systems.
