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Spectral correlations are a powerful tool to study the dynamics of quantummany-body systems. For Hermitian Hamiltonians, quantum chaotic motion isrelated to random matrix theory spectral correlations. Based on recent progressin the application of spectral analysis to non-Hermitian quantum systems, weshow that local level statistics, which probe the dynamics around theHeisenberg time, of a non-Hermitian $q$-body Sachdev-Ye-Kitev (nHSYK) modelwith $N$ Majorana fermions, and its chiral and complex-fermion extensions, arealso well described by random matrix theory for $q > 2$, while for $q = 2$,they are given by the equivalent of Poisson statistics. For that comparison, wecombine exact diagonalization numerical techniques with analytical resultsobtained for some of the random matrix spectral observables. Moreover,depending on $q$ and $N$, we identify $19$ out of the $38$ non-Hermitianuniversality classes in the nHSYK model, including those corresponding to thetenfold way. In particular, we realize explicitly $14$ out of the $15$universality classes corresponding to non-pseudo-Hermitian Hamiltonians thatinvolve universal bulk correlations of classes ${\rm AI}^\dagger$ and ${\rmAII}^\dagger$, beyond the Ginibre ensembles. These results provide strongevidence of striking universal features in non-unitary many-body quantum chaos,which in all cases can be captured by nHSYK models with $q > 2$.

Symmetry Classification and Universality in Non-Hermitian Many-Body Quantum Chaos by the Sachdev-Ye-Kitaev Model