  We investigate structural properties of the completely positive semidefinite
cone $\mathcal{CS}_+^n$, consisting of all the $n \times n$ symmetric matrices
that admit a Gram representation by positive semidefinite matrices of any size.
This cone has been introduced to model quantum graph parameters as conic
optimization problems. Recently it has also been used to characterize the set
$\mathcal Q$ of bipartite quantum correlations, as projection of an affine
section of it. We have two main results concerning the structure of the
completely positive semidefinite cone, namely about its interior and about its
closure. On the one hand we construct a hierarchy of polyhedral cones which
covers the interior of $\mathcal{CS}_+^n$, which we use for computing some
variants of the quantum chromatic number by way of a linear program. On the
other hand we give an explicit description of the closure of the completely
positive semidefinite cone, by showing that it consists of all matrices
admitting a Gram representation in the tracial ultraproduct of matrix algebras.
