  A strong direct product theorem (SDPT) states that solving n instances of a
problem requires Omega(n) times the resources for a single instance, even to
achieve success probability exp(-Omega(n)). We prove that quantum communication
complexity obeys an SDPT whenever the communication lower bound for a single
instance is proved by the generalized discrepancy method, the strongest
technique in that model. We prove that quantum query complexity obeys an SDPT
whenever the query lower bound for a single instance is proved by the
polynomial method, one of the two main techniques in that model. In both
models, we prove the corresponding XOR lemmas and threshold direct product
theorems.
