  We employ the k-th nearest-neighbor estimator of configurational entropy in
order to decode within a parameter-free numerical approach the complex
high-order structural correlations in fluxional molecules going beyond the
usual linear, bivariate correlations. This generic entropy-based scheme for
determining many-body correlations is applied to the complex configurational
ensemble of protonated acetylene, a prototype for fluxional molecules featuring
large-amplitude motion. After revealing the importance of high-order
correlations beyond the simple two-coordinate picture for this molecule, we
analyze in detail the evolution of the relevant correlations with temperature
as well as the impact of nuclear quantum effects down to the ultra-low
temperature regime of 1 K. We find that quantum delocalization and zero-point
vibrations significantly reduce all correlations in protonated acetylene in the
deep quantum regime. At intermediate temperatures of approximately 100 to 800
K, a quantum-to-classical cross-over regime is found where classical mechanics
starts to correctly describe trends in the correlations whereas it even
qualitatively fails below 100 K. Finally, a classical description of the nuclei
provides correlations that are in quantitative agreement with the quantum ones
only at temperatures exceeding 1000 K. This data-intensive analysis has been
made possible due to recent developments of machine learning techniques based
on neural network potentials in full dimensionality that allow us to
exhaustively sample both, the classical and quantum ensemble of protonated
acetylene at essentially converged coupled cluster accuracy. The presented
non-parametric analysis of correlations is expected to complement and guide the
analysis of experimental measurements, in particular multi-dimensional
vibrational spectroscopy, by revealing the complex coupling between various
degrees of freedom.
