As blockchain applications become increasingly widespread, there is a rising demand for on-chain data queries. However, existing schemes for on-chain data queries face a challenge between verifiability and efficiency. Queries on blockchain databases can compromise the authenticity of the query results, while schemes that utilize on-chain Authenticated Data Structure (ADS) have lower efficiency. To overcome this limitation, we propose an efficient and verifiable on-chain data query framework EVeCA. In our approach, we free the full nodes from the task of ADS maintenance by delegating it to a limited number of nodes, and full nodes verify the correctness of ADS by using challenge-based authentication scheme instead of reconstructing them, which prevents the service providers from maintaining incorrect ADS with overwhelming probability. By carefully designing the ADS verification scheme, EVeCA achieves higher efficiency while remaining resilient against adaptive attacks. Our framework effectively eliminates the need for on-chain ADS maintenance, and allows full nodes to participate in ADS maintenance in a cost-effective way. We demonstrate the effectiveness of the proposed scheme through security analysis and experimental evaluation. Compared to existing schemes, our approach improves ADS maintenance efficiency by about 20*.