Show that TQBF $\notin$ SPACE$((\log{n})^4)$?
Hint: TQBF is PSPACE-complete, so it "requires" polynomial space. If there were an algorithm to solve TQBF using space $O(\log^4 n)$, you could use it to solve all problems in PSPACE using surprisingly small space. So surprising, in fact, that the space hierarchy theorem says it can't be.