I have a $\text{NEXP}$-hard problem, that can be solved by a $\text{NEXP}^\text{NP}$ algorithm using a single oracle call.So from Hemaspaandra we know it is in $\text{P}^\text{NE}$, giving us$\text{NEXP}$-completeness under Cook-reductions.Are there (more or less) natural problems complete for $\text{P}^\text{NE}$?What do we know about the space between $\text{NEXP}$ and $\text{P}^\text{NE}$?Anything else that might help to sharpen this result (to Karp-reductions)? 
What do we know about $\text{P}^\text{NE}$
cc.complexity theory;reference request;complexity classes;nexp