Running multiple rounds of a BQP computation, without multiple measurements?
The principle of deferred measurement (see https://en.wikipedia.org/wiki/Deferred_Measurement_Principle) tells us that measurements in the middle of a quantum computation can be simulated by using additional quantum gates instead. In particular, instead of measuring a qubit, we can apply a CNOT gate to that qubit and another qubit (which is used only at this point of the computation).