This preprint introduces a two-component notion of entropic complexity for discrete-time stochastic theories, separating the randomness visible at the boundary from the internal reshaping of laws in time. A theory is defined as a probability law on bi-infinite (or semi-infinite) symbol sequences, while an implementation is a Markovian dynamical system with hidden states and an observation map that reproduces the same boundary law. On the implementation side, the paper defines a temporal complexity as the long-time average of a divergence D (μn+1∥μn) D (₍+₁ₙ) D (μn+1∥μn) between successive one-step marginals of the internal state, for a broad class of classical and quantum divergences satisfying positivity, data processing, and a local lower semicontinuity condition. On the boundary side, it defines an interface complexity as the Shannon entropy rate of the observed symbol process. Basic structural properties are established, including existence of (possibly huge) path-space implementations for any theory, monotonicity of temporal complexity under coarse-graining, and a lower semicontinuity result under weak convergence and uniform absolute continuity assumptions. Finite-state ergodic Markov chains and simple driven two-state models are worked out as examples illustrating vanishing or strictly positive temporal complexity, and clarifying the distinction between high boundary randomness and high internal reshaping rate. The framework is then extended to finite-dimensional quantum systems via Umegaki relative entropy, CPTP dynamics, and POVM measurements, with an explicit discussion of measurement protocols and their limitations. Connections to stochastic thermodynamics, computational mechanics, and quantum resource theories are outlined, suggesting temporal complexity as a flexible, divergence-based diagnostic for non-equilibrium discrete-time dynamics beyond specific microscopic models.
Takahashi K (Tue,) studied this question.