of these earlier results.

Theorem \[mainthm1\] says that one can always replace polynomial-size quantum advice by polynomial-size *classical* advice, together with a polynomial-size quantum *witness* (or equivalently, *untrusted* quantum advice).  Indeed, we can *characterize* the class $\mathsf{BQP/qpoly}$, as equal to the subclass of $\mathsf{QMA/poly}$ in which the quantum witness state $\left\vert \psi_{n}\right\rangle $ can only depend on the input length $n$.[^3]

Using Theorem \[mainthm1\], we also obtain several other results for quantum complexity theory:

1.  Without loss of generality, every quantum advice state can be taken to be the ground state of some local Hamiltonian $H$.  (This essentially follows by combining our $\mathsf{BQP/qpoly}\subseteq\mathsf{QMA/poly}$ result with the result of Kitaev that <span style="font-variant:small-caps;">Local Hamiltonians</span> is $\mathsf{QMA}$-complete.)

2.  It is open whether for every local Hamiltonian $H$ on $n$ qubits, there exists a quantum circuit of size $\operatorname*{poly}\left(  n\right) $ that prepares a ground state of $H$. It is easy to show that an affirmative answer would imply $\mathsf{QMA}=\mathsf{QCMA}$. As a consequence of Theorem \[mainthm1\], we can show that an affirmative answer would also imply $\mathsf{BQP/qpoly}=\mathsf{BQP/poly}$—thereby establishing a previously-unknown connection between quantum proofs and quantum advice.

3.  We generalize Theorem \[mainthm1\] to show that $\mathsf{QCMA/qpoly}\subseteq\mathsf{QMA/poly}$.

4.  We also use our new characterization 