Proposition: Equal marginal entropy, different temporal patterns

An independent uniform four-label process and the deterministic cycle $0,1,2,3,0,\dots$, with a uniform initial phase, both have marginal entropy $\log 4$. Their one-step conditional entropies are $\log 4$ and zero, respectively.

Proof. Both have uniform marginals. The independent process retains uniform conditional law. In the cycle, the next label is determined by the current one.