We aim to understand inherent reasons for lower bounds for QBF proof systems, and revisit and compare two previous approaches in this direction.

The first of these relates size lower bounds for strong QBF Frege systems to circuit lower bounds via strategy extraction (Beyersdorff & Pich, LICS'16). Here we show a refined version of strategy extraction and thereby for any QBF proof system obtain a trichotomy for hardness: (1) via circuit lower bounds, (2) via propositional Resolution lower bounds, or (3) `genuine' QBF lower bounds.

The second approach tries to explain QBF lower bounds through quantifier alternations in a system called relaxing QU-Res (Chen, ICALP'16). We prove a strong lower bound for relaxing QU-Res, which at the same time exhibits significant shortcomings of that model. Prompted by this we propose an alternative, improved version, allowing fewer but more flexible oracle queries in proofs. We show that lower bounds in our new model correspond to the trichotomy obtained via strategy extraction.

We strengthen the definition of $\Sigma_k^p$-QU-Res; we remove the restrictions on oracle queries present in the previous version.

We aim to understand inherent reasons for lower bounds for QBF proof systems and revisit and compare two previous approaches in this direction.

The first of these relates size lower bounds for strong QBF Frege systems to circuit lower bounds via strategy extraction (Beyersdorff & Pich, LICS'16). Here we show a refined version of strategy extraction and thereby for any QBF proof system obtain a trichotomy for hardness: (1) via circuit lower bounds, (2) via propositional Resolution lower bounds, or (3) `genuine' QBF lower bounds.

The second approach tries to explain QBF lower bounds through quantifier alternations in a system called relaxing QU-Res (Chen, ICALP'16). We prove a strong lower bound for relaxing QU-Res, which at the same time exhibits significant shortcomings of that model. Prompted by this we propose an alternative, improved version, allowing fewer but more flexible oracle queries in proofs. We show that lower bounds in our new model correspond to the trichotomy obtained via strategy extraction.