Nonstabilizerness of Permutationally Invariant Systems

Typical measures of nonstabilizerness of a system of N qubits require computing 4N expectation values, one for each Pauli string in the Pauli group, over a state of dimension 2N. For permutationally invariant systems, this exponential overhead can be reduced to just O(N³) expectation values on a state with a dimension $O(N)$. We exploit this simplification to study the nonstabilizerness phase transitions of systems with hundreds of qubits.