  This paper considers three variants of quantum interactive proof systems in
which short (meaning logarithmic-length) messages are exchanged between the
prover and verifier. The first variant is one in which the verifier sends a
short message to the prover, and the prover responds with an ordinary, or
polynomial-length, message; the second variant is one in which any number of
messages can be exchanged, but where the combined length of all the messages is
logarithmic; and the third variant is one in which the verifier sends
polynomially many random bits to the prover, who responds with a short quantum
message. We prove that in all of these cases the short messages can be
eliminated without changing the power of the model, so the first variant has
the expressive power of QMA and the second and third variants have the
expressive power of BQP. These facts are proved through the use of quantum
state tomography, along with the finite quantum de Finetti theorem for the
first variant.
