Grammatical Error Correction (GEC) aims to correct writing errors and help language learners improve their writing skills. However, existing GEC models tend to produce spurious corrections or fail to detect lots of errors. The quality estimation model is necessary to ensure learners get accurate GEC results and avoid misleading from poorly corrected sentences. Well-trained GEC models can generate several high-quality hypotheses through decoding, such as beam search, which provide valuable GEC evidence and can be used to evaluate GEC quality. However, existing models neglect the possible GEC evidence from different hypotheses. This paper presents the Neural Verification Network (VERNet) for GEC quality estimation with multiple hypotheses. VERNet establishes interactions among hypotheses with a reasoning graph and conducts two kinds of attention mechanisms to propagate GEC evidence to verify the quality of generated hypotheses. Our experiments on four GEC datasets show that VERNet achieves state-of-the-art grammatical error detection performance, achieves the best quality estimation results, and significantly improves GEC performance by reranking hypotheses. All data and source codes are available at https://github.com/thunlp/VERNet.