  An important question of quantum information is to characterize genuinely
quantum (beyond-Clifford) resources necessary for universal quantum computing.
Here, we use the Pauli spectrum to quantify how magic, beyond Clifford, typical
many-qubit states are. We first present a phenomenological picture of the Pauli
spectrum based on quantum typicality and then confirm it for Haar random
states. We then introduce filtered stabilizer entropy, a magic measure that can
resolve the difference between typical and atypical states. We proceed with the
numerical study of the Pauli spectrum of states created by random circuits as
well as for eigenstates of chaotic Hamiltonians. We find that in both cases the
Pauli spectrum approaches the one of Haar random states, up to exponentially
suppressed tails. Our results underscore differences between typical and
atypical states from the point of view of quantum information.
