Recent progress in the study of the Sachdev-Ye-Kitaev (SYK) model and its variants have attracted renewed attention in the characterization of quantum chaotic dynamics. Experimental realization of the SYK model has been proposed in various atomic and solid condensed matter setups. [1] In order to characterize quantum many-body chaos, we define a simple quantum generalization [2] of the spectrum of finite-time classical Lyapunov exponents. [3] We study its statistical features for the SYK model and find random matrix behavior, which is lost when the model is deformed away from chaos towards integrability [4] by a random two-fermion term. We provide numerical evidences for the SYK model as well as for the XXZ spin chain with random field. References: [1] M. Franz and M. Rozali, “Mimicking black hole event horizons in atomic and solid-state systems”, Nature Reviews Materials 3, 491 (2018). [2] M. Hanada, H. Shimada, and M. Tezuka, “Universality in Chaos: Lyapunov Spectrum and Random Matrix Theory”, Phys. Rev. E 97, 022224 (2018). [3] H. Gharibyan, M. Hanada, B. Swingle, and M. Tezuka, “Quantum Lyapunov Spectrum”, arXiv:1809.01671. [4] A. M. Garcia-Garcia, B. Loureiro, A. Romero-Bermudez, and M. Tezuka, “Chaotic-Integrable Transition in the Sachdev-Ye-Kitaev Model”, Phys. Rev. Lett. 120, 241603 (2018).