For attempts to separate the bounded nondeterminism hierarchy, I think monotone dualization of prime formulas is the most salient topic.

Consider the decision problem

DUAL
Input: two monotone CNF formulas, from which no literals can be removed
Question: is the one formula the dual of the other?

DUAL can be decided with $O((\log\ n)^2/\log\ \log\ n)$ nondeterministic steps. There is also a quasi-polynomial $n^{o(\log\ n)}$ time upper bound for this problem which has stood since 1996.

So DUAL is in co-$\beta_2$P and also doesn't seem to require the full power of this class. On the other hand, DUAL may be a good candidate for a problem that is outside P = co-$\beta_1$P.

This is surveyed in:

• Thomas Eiter, Kazuhisa Makino, and Georg Gottlob, Computational aspects of monotone dualization: A brief survey, DAM 156 2035– 2049, 2008. doi: 10.1016/j.dam.2007.04.017 (preprint)

I am not sure there is more work along these lines. As with many other areas with the potential to separate P from NP, after some promising early progress new ideas now appear to be necessary.

For attempts to separate the bounded nondeterminism hierarchy, I think monotone dualization of prime formulas is the most salient topic.

Consider the decision problem

MONOTONE DUAL
Input: two monotone CNF formulas, from which no literals can be removed
Question: is the one formula the dual of the other?

MONOTONE DUAL can be decided with $O((\log\ n)^2/\log\ \log\ n)$ nondeterministic steps. There is also a quasi-polynomial $n^{o(\log\ n)}$ time upper bound for this problem which has stood since 1996.

So MONOTONE DUAL is in co-$\beta_2$P and also doesn't seem to require the full power of this class. On the other hand, MONOTONE DUAL may be a good candidate for a problem that is outside P = co-$\beta_1$P.

This is surveyed in:

• Thomas Eiter, Kazuhisa Makino, and Georg Gottlob, Computational aspects of monotone dualization: A brief survey, DAM 156 2035– 2049, 2008. doi: 10.1016/j.dam.2007.04.017 (preprint)

I am not sure there is more work along these lines. As with many other areas with the potential to separate P from NP, after some promising early progress new ideas now appear to be necessary.