What hierarchies and/or hierarchy theorems do you know?
The Fourier Hierarchy as defined in "Yaoyun Shi, Quantum and classical tradeoffs."

From the complexity zoo:


  $\mathsf{FH}_k$ is the class of problems solvable by a uniform family of polynomial-size quantum circuits, with $k$ levels of Hadamard gates and all other gates preserving the computational basis.
  
  
  $\mathsf{FH}_0 = \mathsf{P}$
  $\mathsf{FH}_1 = \mathsf{BPP}$
  $\mathsf{FH}_2$ contains factoring because of Kitaev's phase estimation algorithm.
  
  
  It is an open problem to show that the Fourier hierarchy is infinite relative to an oracle (that is, $\mathsf{FH}_k$ is strictly contained in $\mathsf{FH}_{k+1}$).