$\mathsf{fewP}$ ($\mathsf{NP}$ with few witnesses, see the zoo) is one of the important ambiguity-bounded sub-classes of $\mathsf{NP}$. There are interesting natural problems in this class that are believed to be good candidates for problems in $\mathsf{NPI}$-intermediate such as Beltway problem and the Turnpike problem. Informally, $\mathsf{fewP}$ is the class of $\mathsf{NP}$ problems with polynomialy-bounded number of solutions.
$\mathsf{FewP}$ is defined as:
The class of decision problems solvable by an $\mathsf{NP}$ machine such that If the answer is 'no,' then all computation paths reject. If the answer is 'yes,' then at least one path accepts; furthermore, the number of accepting paths is upper-bounded by a polynomial in n, the size of the input.
We known that $\mathsf{P} \subseteq \mathsf{UP} \subseteq \mathsf{fewP}\subseteq \mathsf{NP}$. It is an open problem whether any inclusion is strict. Specifically, $\mathsf{fewP}$ vs. $\mathsf{NP}$ is an open problem. The Complexity Zoo cites oracle results suggesting that all inclusions are strict and that $\mathsf{fewP}$ does not have a Turing-complete set.
Why do we believe that $\mathsf{fewP \ne NP}$?
Is there any unexpected consequence that follows from $\mathsf{fewP=NP}$ such as the collapse of the polynomial hierarchy or disproving any other widely believed complexity-theoretic conjecture?
