Automatic question generation (QG) has shown promise as a source of synthetic training data for question answering (QA). In this paper we ask: Is textual diversity in QG beneficial for downstream QA? Using top-p nucleus sampling to derive samples from a transformer-based question generator, we show that diversity-promoting QG indeed provides better QA training than likelihood maximization approaches such as beam search. We also show that standard QG evaluation metrics such as BLEU, ROUGE and METEOR are inversely correlated with diversity, and propose a diversity-aware intrinsic measure of overall QG quality that correlates well with extrinsic evaluation on QA.