A standard quadratic optimization problem (StQOP) can be formulated as an instance of a linear optimization problem over the cone of completely positive matrices. Using an inner and outer hierarchy of polyhedral approximations of the cone of completely positive matrices, we study the properties of the lower and upper bounds on the optimal value of an StQOP that arise from these approximations. In particular, we give characterizations of instances for which the bounds are exact at a finite level of the hierarchy and of those instances for which the bounds are exact in the limit.