  The spectral form factor (SFF) captures universal spectral fluctuations as
signatures of quantum chaos, and has been instrumental in advancing multiple
frontiers of physics including the studies of black holes and quantum many-body
systems. However, the measurement of SFF in many-body systems is challenging
due to the difficulty in resolving level spacings that become exponentially
small with increasing system size. Here we experimentally measure the SFF to
probe the presence or absence of chaos in quantum many-body systems using a
superconducting quantum processor with a randomized measurement protocol. For a
Floquet chaotic system, we observe signatures of spectral rigidity of random
matrix theory in SFF given by the ramp-plateau behavior. For a Hamiltonian
system, we utilize SFF to distinguish the quantum many-body chaotic phase and
the prethermal many-body localization. We observe the dip-ramp-plateau behavior
of random matrix theory in the chaotic phase, and contrast the scaling of the
plateau time in system size between the many-body chaotic and localized phases.
Furthermore, we probe the eigenstate statistics by measuring a generalization
of the SFF, known as the partial SFF, and observe distinct behaviors in the
purities of the reduced density matrix in the two phases. This work unveils a
new way of extracting the universal signatures of many-body quantum chaos in
quantum devices by probing the correlations in eigenenergies and eigenstates.
