Separating Logspace from Polynomial time It is clear that any problem that is decidable in deterministic logspace ($L$) runs in at most polynomial time ($P$). There is a wealth of complexity classes between $L$ and $P$. Examples include $NL$, $LogCFL$, $NC^i$, $SAC^i$, $AC^i$, $SC^i$. It is widely believed that $L \neq P$.

In one of my blog posts I mentioned two approaches (along with the corresponding conjectures) towards proving $L \neq P$. Both these approaches are based on branching programs and are 20 years apart !! Are there other approaches and/or conjectures towards separating $L$ from $P$ (or) separating any intermediate classes between $L$ and $P$.
Circuit depth lower bounds (equivalently, formula size lower bounds) are probably the most natural approach:  A Super-$\log^2(n)$ depth lower bound for a problem in $\mathsf P$ would separate $\mathsf P$ from $\mathsf L$, and the Karchmer-Wigderson communication complexity technique may be the natural one for that.