Is it NP-Complete to determine if a quadratic program (QP) has multiple solutions?
[Since I could not edit my comment, I repost it here as an answer] 

To determine if a quadratic program (QP) has multiple solutions (so you have to answer "Yes" if the QP has more than one solutions, and "No" if it has one or no solution) can be shown to be NP-complete by modifying the proof on NP-completeness of QP here http://link.springer.com/article/10.1007%2FBF00120662. 

The proof uses a reduction from k-clique to a QP. Now, add a free variable to the QP, so that it has one solution iff it have multiple ones. Thus, the answer to the problem if QP has multiple solutions or not gives the answer to the question if a graph has a clique of size k or not.