  In stochastic modeling, there has been a significant effort towards finding
predictive models that predict a stochastic process' future using minimal
information from its past. Meanwhile, in condensed matter physics, matrix
product states (MPS) are known as a particularly efficient representation of 1D
spin chains. In this Letter, we associate each stochastic process with a
suitable quantum state of a spin chain. We then show that the optimal
predictive model for the process leads directly to an MPS representation of the
associated quantum state. Conversely, MPS methods offer a systematic
construction of the best known quantum predictive models. This connection
allows an improved method for computing the quantum memory needed for
generating optimal predictions. We prove that this memory coincides with the
entanglement of the associated spin chain across the past-future bipartition.
