  For an interacting spatio-temporal lattice system we introduce a formal way
of expressing multi-time correlation functions of local observables located at
the same spatial point with a time state, i.e. a statistical distribution of
configurations observed along a time lattice. Such a time state is defined with
respect to a particular equilibrium state that is invariant under space and
time translations. The concept is developed within the Rule 54 reversible
cellular automaton, for which we explicitly construct a matrix product form of
the time state, with matrices that act on the 3-dimensional auxiliary space. We
use the matrix-product state to express equal-space time-dependent
density-density correlation function, which, for special maximum-entropy values
of equilibrium parameters, agrees with the previous results. Additionally, we
obtain an explicit expression for the probabilities of observing all multi-time
configurations, which enables us to study distributions of times between
consecutive excitations and prove the absence of decoupling of timescales in
the Rule 54 model.
