This paper investigates the relationship between quantum complexity and stabilizer Rényi entropies (SREs) in random quantum circuits and Hamiltonian evolutions. The authors show that SREs, which quantify nonstabilizerness, saturate their maximum value at a critical number of non-Clifford operations, such as T-gates, or at a critical time for Hamiltonian evolution. Around this critical point, SREs exhibit universal behavior, with their derivative crossing at the same point regardless of the number of qubits. This suggests a connection to phase transitions. For random Clifford circuits doped with T-gates, the critical T-gate density is independent of the number of qubits, while for random Hamiltonian evolution, the critical time scales linearly with the number of qubits for α > 1 and is constant for α < 1. This indicates that α-SREs probe different aspects of nonstabilizerness depending on α: α < 1 relates to Clifford simulation complexity, while α > 1 probes the distance to the closest stabilizer state and approximate state certification cost via Pauli measurements. The authors also show that the Pauli spectrum of random evolution can be approximated by two highly concentrated peaks, allowing for the computation of SREs. They introduce a class of random evolution that can be expressed as random Clifford circuits and rotations, where they provide an exact SRE for α = 2. These results open new approaches to characterizing the complexity of quantum systems. The paper highlights that α-SREs with α < 1 and α > 1 probe different aspects of nonstabilizerness. For α < 1, SREs relate to the complexity of Clifford simulation algorithms, while for α > 1, they relate to the fidelity with the closest stabilizer state. The authors also show that the complexity of tasks related to entanglement can be bounded using the stabilizer nullity, which is a measure of nonstabilizerness. The study demonstrates that the critical time and T-gate density for SRE saturation are maximal for α = 2, indicating that 2-SREs hold a special status. This is consistent with the fact that SREs are monotones for α ≥ 2, while they can violate monotonicity for α < 2. The paper concludes that SREs provide a powerful tool for characterizing the complexity of quantum systems.This paper investigates the relationship between quantum complexity and stabilizer Rényi entropies (SREs) in random quantum circuits and Hamiltonian evolutions. The authors show that SREs, which quantify nonstabilizerness, saturate their maximum value at a critical number of non-Clifford operations, such as T-gates, or at a critical time for Hamiltonian evolution. Around this critical point, SREs exhibit universal behavior, with their derivative crossing at the same point regardless of the number of qubits. This suggests a connection to phase transitions. For random Clifford circuits doped with T-gates, the critical T-gate density is independent of the number of qubits, while for random Hamiltonian evolution, the critical time scales linearly with the number of qubits for α > 1 and is constant for α < 1. This indicates that α-SREs probe different aspects of nonstabilizerness depending on α: α < 1 relates to Clifford simulation complexity, while α > 1 probes the distance to the closest stabilizer state and approximate state certification cost via Pauli measurements. The authors also show that the Pauli spectrum of random evolution can be approximated by two highly concentrated peaks, allowing for the computation of SREs. They introduce a class of random evolution that can be expressed as random Clifford circuits and rotations, where they provide an exact SRE for α = 2. These results open new approaches to characterizing the complexity of quantum systems. The paper highlights that α-SREs with α < 1 and α > 1 probe different aspects of nonstabilizerness. For α < 1, SREs relate to the complexity of Clifford simulation algorithms, while for α > 1, they relate to the fidelity with the closest stabilizer state. The authors also show that the complexity of tasks related to entanglement can be bounded using the stabilizer nullity, which is a measure of nonstabilizerness. The study demonstrates that the critical time and T-gate density for SRE saturation are maximal for α = 2, indicating that 2-SREs hold a special status. This is consistent with the fact that SREs are monotones for α ≥ 2, while they can violate monotonicity for α < 2. The paper concludes that SREs provide a powerful tool for characterizing the complexity of quantum systems.