Savitch's theorem shows that NSPACE($S(n)$) $\subseteq$ SPACE($S(n)^2$), which means that nondeterminism can be replaced by more spaces in this situation. Is it known whether nondeterminism can be replaced by advice strings in a similar situation?
Especially, I would like to ask about "L/quasipoly vs NL/poly". If L/quasipoly $\supset$ NL/poly is proved, is this a new result?
(Here, L/quasipoly means (informally) L with quasipolynomial size advice strings.)
Currently, I think that I may be able to prove that polynomial-size nondeterministic branching programs can be simulated by quasipolynomial-size $O(\log n)$-width boolean circuits.