Does requiring uniqueness of valid answers for Merlin limit the power of Arthur-Merlin protocols?
Koiran's paper Hilbert's Nullstellensatz is in the Polynomial Hierarchy provides a public-coin Arthur-Merlin protocol for establishing that a system of $m$ equations on $n$ unknowns has a solution in $\mathbb{C}^n$, contingent on the Generalized Riemann Hypothesis.  Here Merlin finds a prime $p$ with $H(p)=0$ for some random hash $H$, along with a solution $(a_0,a_1,\cdots, a_n)$ to each of the $m$ equations $\bmod p$.

If the system of equations has no solution $\bmod p$, then there will only be a finite number of $p$ modulo which a solution exists.  If the system does have a solution $\bmod p$, then unconditionally there will be an positive density of $p$ with a solution, and the GRH allows that the $p$ with a solution are "equidistributed" in some sense, such that Merlin gets a win.

Although Koiran gives an example of a solvable system where the density of $p$ is exponentially small, Koiran suggests that if the system is solvable in $\mathbb{C}^n$, then in most cases there will be a large number of $p$ (and a large number of $a$); indeed about $1-1/e$ primes should have a solution IIRCC.

Thus, in the above problem, not only does there exist a valid response that Merlin can issue to Arthur for each challenge, but in fact there could be a large number of valid responses.

Distinguishing the easy systems of equations with solutions modulo many $p$ from the hard systems of equations with a few or only one $p$ seems to require determining properties of a Galois group of the system of equations.