  Nonstabilizerness or `magic' is a key resource for quantum computing and a
necessary condition for quantum advantage. Non-Clifford operations turn
stabilizer states into resourceful states, where the amount of
nonstabilizerness is quantified by resource measures such as stabilizer R\'enyi
entropies (SREs). Here, we show that SREs saturate their maximum value at a
critical number of non-Clifford operations. Close to the critical point SREs
show universal behavior. Remarkably, the derivative of the SRE crosses at the
same point independent of the number of qubits and can be rescaled onto a
single curve. We find that the critical point depends non-trivially on R\'enyi
index $\alpha$. For random Clifford circuits doped with T-gates, the critical
T-gate density scales independently of $\alpha$. In contrast, for random
Hamiltonian evolution, the critical time scales linearly with qubit number for
$\alpha>1$, while is a constant for $\alpha<1$. This highlights that
$\alpha$-SREs reveal fundamentally different aspects of nonstabilizerness
depending on $\alpha$: $\alpha$-SREs with $\alpha<1$ relate to Clifford
simulation complexity, while $\alpha>1$ probe the distance to the closest
stabilizer state and approximate state certification cost via Pauli
measurements. As technical contributions, we observe that the Pauli spectrum of
random evolution can be approximated by two highly concentrated peaks which
allows us to compute its SRE. Further, we introduce a class of random evolution
that can be expressed as random Clifford circuits and rotations, where we
provide its exact SRE. Our results opens up new approaches to characterize the
complexity of quantum systems.
