  The minimal memory required to model a given stochastic process - known as
the statistical complexity - is a widely adopted quantifier of structure in
complexity science. Here, we ask if quantum mechanics can fundamentally change
the qualitative behaviour of this measure. We study this question in the
context of the classical Ising spin chain. In this system, the statistical
complexity is known to grow monotonically with temperature. We evaluate the
spin chain's quantum mechanical statistical complexity by explicitly
constructing its provably simplest quantum model, and demonstrate that this
measure exhibits drastically different behaviour: it rises to a maximum at some
finite temperature then tends back towards zero for higher temperatures. This
demonstrates how complexity, as captured by the amount of memory required to
model a process, can exhibit radically different behaviour when quantum
processing is allowed.
