Efficient Quantum Algorithms for Shifted Quadratic Character Problems
  We introduce the Shifted Legendre Symbol Problem and some variants along with efficient quantum algorithms to solve them. The problems and their algorithms are different from previous work on quantum computation in that they do not appear to fit into the framework of the Hidden Subgroup Problem. The classical complexity of the problem is unknown despite the various results on the irregularity of Legendre Sequences. 
Wim van Dam (Berkeley, CWI), Sean Hallgren (MSRI)