1) What classes contain $\textbf{PSPACE}$, are contained in $\textbf{EXP}$ and (presumably) are not equal to $\textbf{PSPACE}$ nor to $\textbf{EXP}$?
A possible class satisfying this requirement: the class of languages that can be recognized in exponential time and subexponential memory.
2) Which of these classes are provably strictly larger than $\textbf{P}$?