Classical chaos is characterized by the existence of positive Lyapunov exponents (LEs), leading to exponential separation of phase space trajectories with slightly different initial conditions. In the quantum domain, out-of-time ordered correlators (OTOCs) naturally emerge as a probe of quantum chaos, akin to the classical LEs. We probe the OTOCs in a system that disobeys the celebrated Kolmogorov-Arnold-Moser (KAM) theorem using rigorous numerical and mathematical techniques. We consider a quantized kicked harmonic oscillator (KHO) for this purpose. As the harmonic oscillator is a non-KAM integrable system, an extremely small time-dependent perturbation could drive the system chaotic. In particular, at specific points in the parameter space called resonances, the system admits large-scale structural changes to the variations in the parameters. At resonances, we observe that the long-time dynamics of the OTOCs are sensitive to these structural changes, where they grow quadratically as opposed to linear or stagnant growth at non-resonances. On the other hand, our findings suggest that the short-time dynamics remain relatively more stable and show the exponential growth found in the literature for unstable fixed points. We further analytically extract the “quantum Lyapunov exponent” from the OTOC for a special case and show the correspondence with the classical LE. We shall also discuss possible applications of such non-KAM system dynamics in quantum sensing. We then extend our OTOC studies to a finite-dimensional system known as kicked coupled top, having spin degrees of freedom. This system conserves total magnetization along the z-direction and has a smooth classical limit. A systematic study of OTOCs in this system offers (i) fresh insights into scrambling in mixed-phase space - a domain that has not been comprehensively explored before and (ii) implications of the conserved quantities on the scrambling of information. Finally, complementing the OTOCs to probe quantum chaos, we explore emergent higher-order state designs from quantum states generated by chaotic Hamiltonians. The emergence of designs signals deep thermalization in quantum systems, which lie beyond the purview of the conventional Eigenstate Thermalization Hypothesis. in quantum sensing. We then extend our OTOC studies to a finite-dimensional system known as kicked coupled top, having spin degrees of freedom. This system conserves total magnetization along the z-direction and has a smooth classical limit. A systematic study of OTOCs in this system offers (i) fresh insights into scrambling in mixed-phase space - a domain that has not been comprehensively explored before and (ii) implications of the conserved quantities on the scrambling of information. Finally, complementing the OTOCs to probe quantum chaos, we explore emergent higher-order state designs from quantum states generated by chaotic Hamiltonians. The emergence of designs signals deep thermalization in quantum systems, which lie beyond the purview of the conventional Eigenstate Thermalization Hypothesis.