# Is Quasi-polynomial time in PSPACE?

I had done some search on this but I was not able to find an answer either way.

Huck answered it fully. Thanks :)

• Can you move your "comment inside the question" to an actual comment. – Suresh Venkat Oct 12 '11 at 22:56
• @Suresh, I do not think there is enough room for it? I am not sure. – Tayfun Pay Oct 12 '11 at 22:59
• Can you perhaps remove the "comment" part entirely? I do not think that it is appropriate. – Jukka Suomela Oct 12 '11 at 23:14
• Put whatever you can and remove the rest. And post this in Huck's answer. It's not appropriate to insert an answer comment inside the original question – Suresh Venkat Oct 12 '11 at 23:29

Assume $QP \subseteq PSPACE$. Then we have $P \subsetneq QP \subseteq PSPACE$, where the first inclusion is proper by the time hierarchy theorem.
This separates $P$ from $PSPACE$, which is not known to hold, so $QP \subseteq PSPACE$ must also not be known to hold.
Indeed we have that $PSPACE \subseteq QP \Rightarrow PSPACE \subsetneq EXP$, but $QP \nsubseteq PSPACE$ does not separate the two classes by the THT (as stated in the question).