  We investigate query-to-communication lifting theorems for models related to
the quantum adversary bounds. Our results are as follows:
  1. We show that the classical adversary bound lifts to a lower bound on
randomized communication complexity with a constant-sized gadget. We also show
that the classical adversary bound is a strictly stronger lower bound technique
than the previously-lifted measure known as critical block sensitivity, making
our lifting theorem one of the strongest lifting theorems for randomized
communication complexity using a constant-sized gadget.
  2. Turning to quantum models, we show a connection between lifting theorems
for quantum adversary bounds and secure 2-party quantum computation in a
certain "honest-but-curious" model. Under the assumption that such secure
2-party computation is impossible, we show that a simplified version of the
positive-weight adversary bound lifts to a quantum communication lower bound
using a constant-sized gadget. We also give an unconditional lifting theorem
which lower bounds bounded-round quantum communication protocols.
  3. Finally, we give some new results in query complexity. We show that the
classical adversary and the positive-weight quantum adversary are quadratically
related. We also show that the positive-weight quantum adversary is never
larger than the square of the approximate degree. Both relations hold even for
partial functions.
