Arguably, the largest class of stochastic processes generated by means of a
finite memory consists of those that are sequences of observations produced by
sequential measurements in a suitable generalized probabilistic theory (GPT).
These are constructed from a finite-dimensional memory evolving under a set of
possible linear maps, and with probabilities of outcomes determined by linear
functions of the memory state. Examples of such models are given by classical
hidden Markov processes, where the memory state is a probability distribution,
and at each step it evolves according to a non-negative matrix, and hidden
quantum Markov processes, where the memory state is a finite dimensional
quantum state, and at each step it evolves according to a completely positive
map. Here we show that the set of processes admitting a finite-dimensional
explanation do not need to be explainable in terms of either classical
probability or quantum mechanics. To wit, we exhibit families of processes that
have a finite-dimensional explanation, defined manifestly by the dynamics of
explicitly given GPT, but that do not admit a quantum, and therefore not even
classical, explanation in finite dimension. Furthermore, we present a family of
quantum processes on qubits and qutrits that do not admit a classical
finite-dimensional realization, which includes examples introduced earlier by
Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite dimensional
Markov chains, and lower bound the size of the memory of a classical model
realizing a noisy version of the qubit processes.Comment: 18 pages, 0 figure