# It is known that $L \subsetneq PH$?

Is it known whether $$Logspace$$ is strictly contained in the polynomial time hierarchy ?

Are there oracles relative to which these classes are equal / distinct ?

• No, this is not known. For all we know, PH may collapse to uniform $\mathrm{AC}^0[6]$. – Emil Jeřábek Feb 18 at 11:30

This is equivalent to $$LOGSPACE≠NP$$ (which is obviously open). The proof of that equivalence relativizes (at least under the usual oracle models).
And there are oracles making $$LOGSPACE = NP$$ (the PSPACE-complete TQBF works) and making them not equal (the oracle separating P from NP works).