  We study unitary property testing, where a quantum algorithm is given query
access to a black-box unitary and has to decide whether it satisfies some
property. In addition to containing the standard quantum query complexity model
(where the unitary encodes a binary string) as a special case, this model
contains "inherently quantum" problems that have no classical analogue.
Characterizing the query complexity of these problems requires new algorithmic
techniques and lower bound methods.
  Our main contribution is a generalized polynomial method for unitary property
testing problems. By leveraging connections with invariant theory, we apply
this method to obtain lower bounds on problems such as determining recurrence
times of unitaries, approximating the dimension of a marked subspace, and
approximating the entanglement entropy of a marked state. We also present a
unitary property testing-based approach towards an oracle separation between
$\mathsf{QMA}$ and $\mathsf{QMA(2)}$, a long standing question in quantum
complexity theory.
