  Effective and efficient forecasting relies on identification of the relevant
information contained in past observations -- the predictive features -- and
isolating it from the rest. When the future of a process bears a strong
dependence on its behaviour far into the past, there are many such features to
store, necessitating complex models with extensive memories. Here, we highlight
a family of stochastic processes whose minimal classical models must devote
unboundedly many bits to tracking the past. For this family, we identify
quantum models of equal accuracy that can store all relevant information within
a single two-dimensional quantum system (qubit). This represents the ultimate
limit of quantum compression and highlights an immense practical advantage of
quantum technologies for the forecasting and simulation of complex systems.
