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Motivated by the Gottesman-Knill theorem, we present a detailed study of the quantum complexity of $p$-shell and $sd$-shell nuclei. Valence-space nuclear shell-model wavefunctions generated by the BIGSTICK code are mapped to qubit registers using the Jordan-Wigner mapping (12 qubits for the $p$-shell and 24 qubits for the $sd$-shell), from which measures of the many-body entanglement ($n$-tangles) and magic (non-stabilizerness) are determined. While exact evaluations of these measures are possible for nuclei with a modest number of active nucleons, Monte Carlo simulations are required for the more complex nuclei. The broadly-applicable Pauli-String $IZ$ exact (PSIZe-) MCMC technique is introduced to accelerate the evaluation of measures of magic in deformed nuclei (with hierarchical wavefunctions), by factors of $\sim 8$ for some nuclei. Significant multi-nucleon entanglement is found in the $sd$-shell, dominated by proton-neutron configurations, along with significant measures of magic. This is evident not only for the deformed states, but also for nuclei on the path to instability via regions of shape coexistence and level inversion. These results indicate that quantum-computing resources will accelerate precision simulations of such nuclei and beyond., Comment: 36 pages, 13 figures