Complexity Theory Consequences of $\mathsf{NP} = \mathsf{QP}$
We can prove $\mathrm{NP\ne QP}$ unconditionally: $\mathrm{QP}$ is closed under quasipolynomial-time reductions, whereas $\mathrm{NP}$ is not (a simple padding argument shows that the closure of $\mathrm{NP}$ under quasipolynomial-time reductions includes all languages computable in nondeterministic quasipolynomial time, and this class strictly contains $\mathrm{NP}$ by the nondeterministic time-hierarchy theorem).
As for consequences of the weaker assumption $\mathrm{NP\subseteq QP}$, see the question Is NP in $DTIME(n^{poly\log n})$?.