  In this paper we study the well-known Chv\'atal-Gomory (CG) procedure for the
class of integer semidefinite programs (ISDPs). We prove several results
regarding the hierarchy of relaxations obtained by iterating this procedure. We
also study different formulations of the elementary closure of spectrahedra. A
polyhedral description of the elementary closure for a specific type of
spectrahedra is derived by exploiting total dual integrality for SDPs.
Moreover, we show how to exploit (strengthened) CG cuts in a branch-and-cut
framework for ISDPs. Different from existing algorithms in the literature, the
separation routine in our approach exploits both the semidefinite and the
integrality constraints. We provide separation routines for several common
classes of binary SDPs resulting from combinatorial optimization problems. In
the second part of the paper we present a comprehensive application of our
approach to the quadratic traveling salesman problem (QTSP). Based on the
algebraic connectivity of the directed Hamiltonian cycle, two ISDPs that model
the QTSP are introduced. We show that the CG cuts resulting from these
formulations contain several well-known families of cutting planes. Numerical
results illustrate the practical strength of the CG cuts in our branch-and-cut
algorithm, which outperforms alternative ISDP solvers and is able to solve
large QTSP instances to optimality.
