We consider the power of various quantum complexity classes with the
restriction that states and operators are defined over a real, rather than
complex, Hilbert space. It is well know that a quantum circuit over the complex
numbers can be transformed into a quantum circuit over the real numbers with
the addition of a single qubit. This implies that BQP retains its power when
restricted to using states and operations over the reals. We show that the same
is true for QMA(k), QIP(k), QMIP, and QSZK.Comment: Significant improvements from previous version, in particular showing
both containments (eg. QMA_R is in QMA and vice versa