  The complexity class QMA is the quantum analog of the classical complexity
class NP. The functional analogs of NP and QMA, called functional NP (FNP) and
functional QMA (FQMA), consist in either outputting a (classical or quantum)
witness, or outputting NO if there does not exist a witness.The classical
complexity class Total Functional NP (TFNP) is the subset of FNP for which it
can be shown that the NO outcome never occurs. TFNP includes many natural and
important problems. In the present work we introduce the complexity class of
Total Functional QMA (TFQMA), the quantum analog of TFNP. We show that FQMA and
TFQMA can be defined in such a way that they do not depend on the values of the
completeness and soundness probabilities. In so doing we introduce new notions
such as the eigenbasis and spectrum of a quantum verification procedure which
are of interest by themselves. We then provide examples of problems that lie in
TFQMA, coming from areas such as the complexity of k-local Hamiltonians and
public key quantum money. In the context of black-box groups, we note that
Group Non-Membership, which was known to belong to QMA, in fact belongs to
TFQMA. We also provide a simple oracle with respect to which we have a
separation between FBQP and TFQMA. In the conclusion we discuss the relation
between TFQMA, public key quantum money, and the complexity of quantum states.
