The paper explores the spread and spectral complexity in quantum systems transitioning from integrability to chaos, focusing on the mixed-field Ising model and the next-to-nearest-neighbor (NNN) deformation of the Heisenberg XXZ spin chain. The authors verify that the presence of a peak in spread complexity before its saturation is a characteristic feature of chaotic systems. They find that the saturation value of spread complexity depends on both the spectral statistics of the Hamiltonian and the specific state, but there is a universal bound determined by the symmetries and dimension of the Hamiltonian, which is realized by the thermofield double (TFD) state at infinite temperature. The time scales at which spread complexity and spectral form factor change behavior are found to be consistent and independent of chaotic properties. For spectral complexity, the key factor determining its saturation value and timescale in chaotic systems is the minimum energy difference in the spectrum, explaining earlier saturation compared to integrable systems. The paper concludes by discussing the properties of the TFD and its potential for probing chaos in quantum many-body systems.The paper explores the spread and spectral complexity in quantum systems transitioning from integrability to chaos, focusing on the mixed-field Ising model and the next-to-nearest-neighbor (NNN) deformation of the Heisenberg XXZ spin chain. The authors verify that the presence of a peak in spread complexity before its saturation is a characteristic feature of chaotic systems. They find that the saturation value of spread complexity depends on both the spectral statistics of the Hamiltonian and the specific state, but there is a universal bound determined by the symmetries and dimension of the Hamiltonian, which is realized by the thermofield double (TFD) state at infinite temperature. The time scales at which spread complexity and spectral form factor change behavior are found to be consistent and independent of chaotic properties. For spectral complexity, the key factor determining its saturation value and timescale in chaotic systems is the minimum energy difference in the spectrum, explaining earlier saturation compared to integrable systems. The paper concludes by discussing the properties of the TFD and its potential for probing chaos in quantum many-body systems.