In this talk we look on conic reformulations of non-convex optimization problems over some closed set K with a quadratic objective function and with additional linear and binary constraints. We provide a detailed study of two different situations where such an optimization problem can be reformulated as a linear optimization problem over the dual cone of the cone of (1XK)-semidefinite matrices. Thereby, a matrix M is denoted set-semidefinite w.r.t. some set S if the quadratic form of the matrix is nonnegative on S. The first situation, which assures such a reformulation, requires the boundedness of the optimal value of the conic problem while the second situation requires some assumption on the asymptotic cone of K. These assumptions are for instance satisfied if K is a bounded or convex. They are also satisfied if K is defined by one, possibly non-convex, quadratic inequality. This is a generalization of the completely positive representation results by Burer for K the nonnegative orthant [Mathematical programming 2009] and connects to a result originally provided by Sturm and Zhang [Math. Oper. Res. 2003].