Two fundamental generalizations of Gorbunov’s and Pinsker’s nonanticipatory ϵ−entropy are formulated and analyzed for a tuple of partially observable, finite-dimensional, random processes (Xn,Yn), where ${X^n} \triangleq \left\{ {{X_1}, \ldots ,{X_n}} \right\}$ is the unobserved state process and Yn ≜ {Y1,…,Yn} is the observable process-a noisy version of Xn, subject to a fidelity between Xn, and its reproduction ${\hat X^n} \triangleq \left\{ {{{\hat X}_1}, \ldots ,{{\hat X}_n}} \right\}$. The encoder observes causally Yn and past reproductions ${\hat X^n}$ may or may not be available to both the encoder and the decoder. Theorem 1 gives a tight lower bound on the operational rate of zero-delay codes, when ${\hat X^n}$ is causally available to the decoder only, in terms of a state-dependent nonanticipatory ϵ−entropy of a state process Zn, which is fundamentally different from a corresponding nonanticipatory ϵ−entropy, when ${\hat X^n}$ is causally available to both the encoder and the decoder. Theorem 2 identifies sufficient conditions for the two nonanticipatory ϵ−entropies to coincide. Theorem 3 identifies the information structure of the optimal test-channel distributions. The paper also discusses applications to jointly Gaussian partially observable processes (Xn,Yn) with a square-error fidelity criterion, and derives characterizations of the two nonanticipatory ϵ−entropies.