  The Pauli measurements (the measurements that can be performed with Clifford
operators followed by measurement in the computational basis) are a fundamental
object in quantum information. It is well-known that there is no assignment of
outcomes to all Pauli measurements that is both complete and consistent. We
define two classes of hidden variable assignments based on relaxing either
condition. Partial hidden variable assignments retain the consistency
condition, but forfeit completeness. Contextual hidden variable assignments
retain completeness but forfeit consistency. We use techniques from spectral
graph theory to characterize the incompleteness and inconsistency of the
respective hidden variable assignments. As an application, we interpret our
incompleteness result as a statement of contextuality and our inconsistency
result as a statement of nonlocality. Our results show that we can obtain large
amounts of contextuality and nonlocality using Clifford gates and measurements.
