Consider the decision problem that when given a graph, we need to decide if a particular edge belongs to any optimal solution to the traveling salesman problem on that graph.
It may be argued that the complexity of this problem is strictly greater than any co-NP problem. The idea is that it's perhaps impossible to come up with a counter-example, since we need e.g. an optimal tour candidate before we can consider a counterexample (but no optimal tour candidate is given in this problem statement).
On the other hand, it may be argued that the complexity of this problem is strictly smaller than any P-space-complete problem, as our problem may be seen as
$\exists$"a tour A containing x" $\forall$"tours B": (some formula stating that A <= B)
whereas some probably "minimal" P-space-complete problem has O(n) alternations of $\exists$ and $\forall$: the quantified boolean formula problem (QBF).
Based on this argument, is it reasonable to expect that there's a complexity class between co-NP and PSPACE? Does this particular class have a name? Can we expect to find arbitrarily more such classes by adding another one alternation of $\forall$ or $\exists$ to the previously found such class?