# What is consequence of $PH\subseteq NSPACE((\log n)^2)$?

What is consequences of $PH\subseteq NSPACE((\log n)^2)$?

We don't even know PH is equals to L or not. I am wondering what will be happened when $PH\subseteq NSPACE((\log n)^2)$?

• For $NP \subseteq NSPACE((\log(n)^2)$, it contredicts the ETH, since it implies that all $NP$ problem can be solve in $n^{\log(n)}$ times. From cstheory.stackexchange.com/questions/36750/… you get that Exp-time hiearchy collapse at the second level. – C.P. Jul 28 '18 at 10:31