To clarify, there are public-key cryptosystems with no known quantum attacks, but it's not known how to build such systems based on problems that are known to be QMA-complete (or NP-complete).
> Currently, our best candidates for such trapdoor OWF's are based on lattice problems, like the Shortest Vector Problem (SVP) that I described earlier. Whereas factoring reduces to the abelian hidden subgroup problem, which is solvable in quantum polynomial time, SVP is only known to reduce to the dihedral hidden subgroup problem, which is not known to be solvable in quantum polynomial time despite a decade of effort.
> Inspired by this observation, and building on earlier work by Ajtai and Dwork, Oded Regev has recently proposed public-key cryptosystems that are provably secure against quantum eavesdroppers, assuming SVP is hard for quantum computers.
Here's a discussion from Scott Aaronson (which has a corresponding chapter in his new book): http://www.scottaaronson.com/democritus/lec8.html
> Currently, our best candidates for such trapdoor OWF's are based on lattice problems, like the Shortest Vector Problem (SVP) that I described earlier. Whereas factoring reduces to the abelian hidden subgroup problem, which is solvable in quantum polynomial time, SVP is only known to reduce to the dihedral hidden subgroup problem, which is not known to be solvable in quantum polynomial time despite a decade of effort.
> Inspired by this observation, and building on earlier work by Ajtai and Dwork, Oded Regev has recently proposed public-key cryptosystems that are provably secure against quantum eavesdroppers, assuming SVP is hard for quantum computers.