  Controllable arrays of ions and ultra-cold atoms can simulate complex
many-body phenomena and may provide insights into unsolved problems in modern
science. To this end, experimentally feasible protocols for quantifying the
buildup of quantum correlations and coherence are needed, as performing full
state tomography does not scale favorably with the number of particles. Here we
develop and experimentally demonstrate such a protocol, which uses time
reversal of the many-body dynamics to measure out-of-time-order correlation
functions (OTOCs) in a long-range Ising spin quantum simulator with more than
100 ions in a Penning trap. By measuring a family of OTOCs as a function of a
tunable parameter we obtain fine-grained information about the state of the
system encoded in the multiple quantum coherence spectrum, extract the quantum
state purity, and demonstrate the buildup of up to 8-body correlations. Future
applications of this protocol could enable studies of many-body localization,
quantum phase transitions, and tests of the holographic duality between quantum
and gravitational systems.
