  We establish that there are properties of quantum many-body dynamics which
are efficiently learnable if we are given access to out-of-time-order
correlators (OTOCs), but which require exponentially many operations in the
system size if we can only measure time-ordered correlators. This implies that
any experimental protocol which reconstructs OTOCs solely from time-ordered
correlators must be, in certain cases, exponentially inefficient. Our proofs
leverage and generalize recent techniques in quantum learning theory. Along the
way, we elucidate a general definition of time-ordered versus out-of-time-order
experimental measurement protocols, which can be considered as classes of
adaptive quantum learning algorithms. Moreover, our results provide a
theoretical foundation for novel applications of OTOCs in quantum simulations.

Quantum Physics;High Energy Physics - Theory