  The spectral form factor (SFF), characterizing statistics of energy
eigenvalues, is a key diagnostic of many-body quantum chaos. In addition,
partial spectral form factors (PSFFs) can be defined which refer to subsystems
of the many-body system. They provide unique insights into energy eigenstate
statistics of many-body systems, as we show in an analysis on the basis of
random matrix theory and of the eigenstate thermalization hypothesis. We
propose a protocol that allows the measurement of the SFF and PSFFs in quantum
many-body spin models, within the framework of randomized measurements. Aimed
to probe dynamical properties of quantum many-body systems, our scheme employs
statistical correlations of local random operations which are applied at
different times in a single experiment. Our protocol provides a unified testbed
to probe many-body quantum chaotic behavior, thermalization and many-body
localization in closed quantum systems which we illustrate with numerical
simulations for Hamiltonian and Floquet many-body spin-systems.
