  The Quadratic Assignment Problem (QAP) is a well-known NP-hard problem that
is equivalent to optimizing a linear objective function over the QAP polytope.
The QAP polytope with parameter $n$ - \qappolytope{n} - is defined as the
convex hull of rank-$1$ matrices $xx^T$ with $x$ as the vectorized $n\times n$
permutation matrices.
  In this paper we consider all the known exponential-sized families of
facet-defining inequalities of the QAP-polytope. We describe a new family of
valid inequalities that we show to be facet-defining. We also show that
membership testing (and hence optimizing) over some of the known classes of
inequalities is coNP-complete. We complement our hardness results by showing a
lower bound of $2^{\Omega(n)}$ on the extension complexity of all relaxations
of \qappolytope{n} for which any of the known classes of inequalities are
valid.
