I had done some search on this but I was not able to find an answer either way.
Huck answered it fully. Thanks :)
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Here's a simple argument that shows that QP is not known to be in PSPACE:
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).