The Quadratic Assignment Problem (QAP) consists of finding a permutation that minimizes a quadratic objective function. Exact methods generally rely on a branch-and-bound procedure, the efficiency of which depends heavily on the quality of its lower bound. In integer linear programming, several bounds have been investigated, exhibiting different trade-offs between speed and quality. The Gilmore-Lawler bound appears to be the most commonly used in practice. It requires solving a linear assignment problem (LAP) for each variable-value pair. We show how to obtain this bound using Singleton Node Consistency (SNC) and LAP. In Cost Function Networks (CFNs), we propose a reformulation that transforms the result of applying LAP to a given variable-value pair into cost functions of arity 1 and 2, which can be added to the original problem. Combined with existing lower bounds for CFNs, including EDAC and a recent CFN propagator for AllDifferent, this method (SNC-LAP-GLB), used as a preprocessing, significantly increases the initial lower bound and accelerates the search, resulting in competitive results on the QAPLIB benchmark. We then propose an extension of the AllDifferent propagator for the Global Cardinality Constraint. It allows us to exploit variable symmetries on some challenging QAPLIB instances, thus improving the results.