Containment of NP and Σ2^P within PMA

Ascertain whether the class PMA (Polynomial Majority Argument), defined via the Edge-Majority framework and known to satisfy coNP ⊆ PMA ⊆ S_2^P, contains NP and whether it contains Σ2^P. In particular, determine if NP ⊆ PMA and if Σ2^P ⊆ PMA.

Background

PMA is introduced as a syntactic subclass of UΣ2 based on a majority-edge accounting principle in bipartite graphs with polynomial-time edge queries and an enforced bound on the number of admissible edge labels. The authors prove coNP ⊆ PMA and PMA ⊆ S_2^P, giving both lower and upper bounds.

Unlike PTW (Polynomial Tournament Winner), where they establish Σ2^P ⊆ PTW, the authors cannot currently position PMA relative to NP or Σ2^P and explicitly note this uncertainty.