  Attacks on classical cryptographic protocols are usually modeled by allowing
an adversary to ask queries from an oracle. Security is then defined by
requiring that as long as the queries satisfy some constraint, there is some
problem the adversary cannot solve, such as compute a certain piece of
information. In this paper, we introduce a fundamentally new model of quantum
attacks on classical cryptographic protocols, where the adversary is allowed to
ask several classical queries in quantum superposition. This is a strictly
stronger attack than the standard one, and we consider the security of several
primitives in this model. We show that a secret-sharing scheme that is secure
with threshold $t$ in the standard model is secure against superposition
attacks if and only if the threshold is lowered to $t/2$. We use this result to
give zero-knowledge proofs for all of NP in the common reference string model.
While our protocol is classical, it is sound against a cheating unbounded
quantum prover and computational zero-knowledge even if the verifier is allowed
a superposition attack. Finally, we consider multiparty computation and show
that for the most general type of attack, simulation based security is not
possible. However, putting a natural constraint on the adversary, we show a
non-trivial example of a protocol that can indeed be simulated.
