  Identifying and extracting the past information relevant to the future
behaviour of stochastic processes is a central task in the quantitative
sciences. Quantum models offer a promising approach to this, allowing for
accurate simulation of future trajectories whilst using less past information
than any classical counterpart. Here we introduce a class of phase-enhanced
quantum models, representing the most general means of causal simulation with a
unitary quantum circuit. We show that the resulting constructions can display
advantages over previous state-of-art methods - both in the amount of
information they need to store about the past, and in the minimal memory
dimension they require to store this information. Moreover, we find that these
two features are generally competing factors in optimisation - leading to an
ambiguity in what constitutes the optimal model - a phenomenon that does not
manifest classically. Our results thus simultaneously offer new quantum
advantages for stochastic simulation, and illustrate further qualitative
differences in behaviour between classical and quantum notions of complexity.
