  There is a widespread perception that dynamical evolution of integrable
systems should be simpler in a quantifiable sense than the evolution of generic
systems, though demonstrating this relation between integrability and reduced
complexity in practice has remained elusive. We provide a connection of this
sort by constructing a specific matrix in terms of the eigenvectors of a given
quantum Hamiltonian. The null eigenvalues of this matrix are in one-to-one
correspondence with conserved quantities that have simple locality properties
(a hallmark of integrability). The typical magnitude of the eigenvalues, on the
other hand, controls an explicit bound on Nielsen's complexity of the quantum
evolution operator, defined in terms of the same locality specifications. We
demonstrate how this connection works in a few concrete examples of quantum
spin chains that possess diverse arrays of highly structured conservation laws
mandated by integrability.
