Many verification and synthesis approaches rely on solving techniques for quantified Boolean formulas (QBF). Consequently, solution witnesses, in the form of Boolean functions, become more and more important as they represent implementations or counterexamples. We present a recursive counterexample guided abstraction and refinement algorithm (CEGAR) for solving and certifying QBFs that exploits structural reasoning on the formula level. The algorithm decomposes the given QBF into one

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... al formula for every block of quantifiers that abstracts from assignments of variables not bound by this quantifier block. Further, we show how to derive an efficient certification extraction method on top of the algorithm. We report on experimental evaluation of this algorithm in the solver QuAbS (Quantified Abstraction Solver) which won the most recent QBF competition (QBFEVAL'18). Further, we show the effectiveness of the certification approach using synthesis benchmarks and a case study for synthesizing winning strategies in Petri Games.