Is there a BQP algorithm for each level of the polynomial hierarchy PH?

Note: Cross-posted on Theoretical Computer Science SE.

This question is inspired by thinking about quantum computing power with respect to games, such as chess/checkers/other toy games. Games fit naturally into the polynomial hierarchy $$\mathrm{PH}$$; I'm curious about follow-up questions.

Every Venn diagram or Hasse diagram I see illustrating the "standard model" of computational complexity describes a universe of $$\mathrm{PSPACE}$$ problems, and puts $$\mathrm{BQP}$$ into a position containing all of $$\mathrm{P}$$, and not containing all of $$\mathrm{NP}$$, but cutting through to outside of the polynomial hierarchy $$\mathrm{PH}$$. That is, such Venn diagrams posit that there are likely problems efficiently solvable with a quantum computer that are outside of $$\mathrm{PH}$$.

But how does this "cut through?"

That is, does this imply that there must be a $$\mathrm{BQP}$$ problem in the first level of the hierarchy, one in the second level of the hierarchy, one in the third level $$\cdots$$, and one such as "forrelation" (correlation of Fourier series) completely outside of the hierarchy?

Or could it be that there are some $$\mathrm{BQP}$$ problems in the first level of the hierarchy, some outside of the hierarchy, and an infinite number of levels of the hierarchy that are voided of any $$\mathrm{BQP}$$ problems?

See, e.g., the above picture from the Quanta Magazine article "Finally, a Problem that Only Quantum Computers Will Ever Be Able to Solve" link. Could $$\mathrm{BQP}$$ be disconnected between $$\mathrm{NP}$$ and the island outside of $$\mathrm{PH}$$? Or must there be a bridge over $$\mathrm{PH}$$ connecting the two?