  How much information do we need about a process' past to faithfully simulate
its future? The statistical complexity is a prominent quantifier of structure
for stochastic processes. Quantum machines, however, can simulate classical
stochastic processes while storing significantly less information than their
optimal classical counterparts. This implies qualitative divergences between
classical and quantum statistical complexity. Here, we develop error-tolerant
techniques to witness such divergences, enabling us to account for the
inevitable imperfections in realising quantum stochastic simulators with
present-day quantum technology. We apply these tools to experimentally verify
the quantum memory advantage in simulating an Ising spin chain, even when
accounting for experimental distortion. This then leads us to observe a
recently conjectured effect, the ambiguity of simplicity$\unicode{x2013}$the
notion that the relative complexity of two different processes can depend on
whether we model the process using classical or quantum means of information
processing.
