  We study a model where two opposing provers debate over the membership status
of a given string in a language, trying to convince a weak verifier whose coins
are visible to all. We show that the incorporation of just two qubits to an
otherwise classical constant-space verifier raises the class of debatable
languages from at most $\mathsf{NP}$ to the collection of all Turing-decidable
languages (recursive languages). When the verifier is further constrained to
make the correct decision with probability 1, the corresponding class goes up
from the regular languages up to at least $\mathsf{E}$. We also show that the
quantum model outperforms its classical counterpart when restricted to run in
polynomial time, and demonstrate some non-context-free languages which have
such short debates with quantum verifiers.
