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Arguably, the largest class of stochastic processes generated by means of afinite memory consists of those that are sequences of observations produced bysequential measurements in a suitable generalized probabilistic theory (GPT).These are constructed from a finite-dimensional memory evolving under a set ofpossible linear maps, and with probabilities of outcomes determined by linearfunctions of the memory state. Examples of such models are given by classicalhidden Markov processes, where the memory state is a probability distribution,and at each step it evolves according to a non-negative matrix, and hiddenquantum Markov processes, where the memory state is a finite dimensionalquantum state, and at each step it evolves according to a completely positivemap. Here we show that the set of processes admitting a finite-dimensionalexplanation do not need to be explainable in terms of either classicalprobability or quantum mechanics. To wit, we exhibit families of processes thathave a finite-dimensional explanation, defined manifestly by the dynamics ofexplicitly given GPT, but that do not admit a quantum, and therefore not evenclassical, explanation in finite dimension. Furthermore, we present a family ofquantum processes on qubits and qutrits that do not admit a classicalfinite-dimensional realization, which includes examples introduced earlier byFox, Rubin, Dharmadikari and Nadkarni as functions of infinite dimensionalMarkov chains, and lower bound the size of the memory of a classical modelrealizing a noisy version of the qubit processes.

Quantum theory in finite dimension cannot explain every general process with finite memory