  The holographic gauge/gravity duality provides an explicit reduction of
quantum field theory (QFT) calculations in the semi-classical large-$N$ limit
to sets of `gravitational' differential equations whose analysis can reveal all
details of the spectra of thermal QFT correlators. We argue that in certain
cases, a complete reconstruction of the spectrum and of the corresponding
correlator is possible from only the knowledge of an infinite, discrete set of
pole-skipping points traversed by a single (hydrodynamic) mode computed in a
series expansion in an inverse number of spacetime dimensions. Conceptually,
this reduces the computation of a QFT correlator spectrum to performing a set
of purely algebraic manipulations. With the help of the pole-skipping analysis,
we also uncover a novel structure underpinning the coefficients that enter the
hydrodynamic dispersion relations.
