  The dynamics of interacting quantum many-body systems has two seemingly
disparate but fundamental facets. The first is the dynamics of real-space local
observables, and if and how they thermalise. The second is to interpret the
dynamics of the many-body state as that of a fictitious particle on the
underlying Hilbert-space graph. In this work, we derive an explicit connection
between these two aspects of the dynamics. We show that the temporal decay of
the autocorrelation in a disordered quantum spin chain is explicitly encoded in
how the return probability on Hilbert space approaches its late-time
saturation. As such, the latter has the same functional form in time as the
decay of autocorrelations but with renormalised parameters. Our analytical
treatment is rooted in an understanding of the morphology of the time-evolving
state on the Hilbert-space graph, and corroborated by exact numerical results.
