The paper investigates the relationship between quantum complexity and the number of non-Clifford operations, a key resource for quantum computing. The authors focus on stabilizer Rényi entropies (SREs) to quantify nonstabilizerness, which measures the amount of non-Clifford operations needed to transform a stabilizer state into a resourceful state. They find that SREs saturate their maximum value at a critical number of non-Clifford operations, and near this critical point, SREs exhibit universal behavior. The derivative of the SRE crosses at the same point for all numbers of qubits and can be rescaled onto a single curve, suggesting a connection to phase transitions. For random Clifford circuits doped with T-gates, the critical T-gate density scales independently of the number of qubits, while for random Hamiltonian evolution, the critical time scales linearly with the number of qubits for \(\alpha > 1\), and is constant for \(\alpha < 1\). This highlights that different values of \(\alpha\) probe different aspects of nonstabilizerness: \(\alpha < 1\) relates to Clifford simulation complexity, while \(\alpha > 1\) probes the distance to the closest stabilizer state and approximate state certification cost via Pauli measurements. The authors also introduce a class of random evolution that can be expressed as random Clifford circuits and rotations, for which they provide an exact expression for the SRE. Their results open new approaches to characterizing the complexity of quantum systems.The paper investigates the relationship between quantum complexity and the number of non-Clifford operations, a key resource for quantum computing. The authors focus on stabilizer Rényi entropies (SREs) to quantify nonstabilizerness, which measures the amount of non-Clifford operations needed to transform a stabilizer state into a resourceful state. They find that SREs saturate their maximum value at a critical number of non-Clifford operations, and near this critical point, SREs exhibit universal behavior. The derivative of the SRE crosses at the same point for all numbers of qubits and can be rescaled onto a single curve, suggesting a connection to phase transitions. For random Clifford circuits doped with T-gates, the critical T-gate density scales independently of the number of qubits, while for random Hamiltonian evolution, the critical time scales linearly with the number of qubits for \(\alpha > 1\), and is constant for \(\alpha < 1\). This highlights that different values of \(\alpha\) probe different aspects of nonstabilizerness: \(\alpha < 1\) relates to Clifford simulation complexity, while \(\alpha > 1\) probes the distance to the closest stabilizer state and approximate state certification cost via Pauli measurements. The authors also introduce a class of random evolution that can be expressed as random Clifford circuits and rotations, for which they provide an exact expression for the SRE. Their results open new approaches to characterizing the complexity of quantum systems.