  Simulations of stochastic processes play an important role in the
quantitative sciences, enabling the characterisation of complex systems. Recent
work has established a quantum advantage in stochastic simulation, leading to
quantum devices that execute a simulation using less memory than possible by
classical means. To realise this advantage it is essential that the memory
register remains coherent, and coherently interacts with the processor,
allowing the simulator to operate over many time steps. Here we report a
multi-time-step experimental simulation of a stochastic process using less
memory than the classical limit. A key feature of the photonic quantum
information processor is that it creates a quantum superposition of all
possible future trajectories that the system can evolve into. This
superposition allows us to introduce, and demonstrate, the idea of comparing
statistical futures of two classical processes via quantum interference. We
demonstrate interference of two 16-dimensional quantum states, representing
statistical futures of our process, with a visibility of 0.96 $\pm$ 0.02.
