We initiate an in-depth proof-complexity analysis of polynomial calculus (Q-PC) for Quantified Boolean Formulas (QBF). In the course of this we establish a tight proof-size characterisation of Q-PC in terms of a suitable circuit model (polynomial decision lists). Using this correspondence we show a size-degree relation for Q-PC, similar in spirit, yet different from the classic size-degree formula for propositional PC by Impagliazzo, Pudlák and Sgall (1999).
We use the circuit characterisation together with the size-degree relation to obtain various new lower bounds on proof size in Q-PC. This leads to incomparability results for Q-PC systems over different fields.