  We explore correlations of eigenstates around the many-body localization
(MBL) transition in their dependence on the energy difference (frequency)
$\omega$ and disorder $W$. In addition to the genuine many-body problem, XXZ
spin chain in random field, we consider localization on random regular graphs
(RRG) that serves as a toy model of the MBL transition. Both models show a very
similar behavior. On the localized side of the transition, the eigenstate
correlation function $\beta(\omega)$ shows a power-law enhancement of
correlations with lowering $\omega$; the corresponding exponent depends on $W$.
The correlation between adjacent-in-energy eigenstates exhibits a maximum at
the transition point $W_c$, visualizing the drift of $W_c$ with increasing
system size towards its thermodynamic-limit value. The correlation function
$\beta(\omega)$ is related, via Fourier transformation, to the Hilbert-space
return probability. We discuss measurement of such (and related) eigenstate
correlation functions on state-of-the-art quantum computers and simulators.
