## 3 Problem Statement  
Spatio-temporal big data is a complex and rich data set that contains a large amount of information related to time and space and has important application value in many fields. However, the storage of spatio-temporal big data faces the following specific challenges: First, the data volume is huge, including a large amount of geographic information, sensor data, and real-time data, resulting in low storage and processing efficiency; second, insufficient scalability, rapid data growth, and difficulty in meeting high scalability storage requirements; in addition, real-time processing and access are required, but the synchronization and verification speed of on-chain storage is slow, making it difficult to meet real-time requirements; at the same time, data privacy and security issues cannot be ignored, and the public storage of sensitive information will bring risks of privacy leakage. In addition, spatio-temporal big data types are diverse, including text, images, videos, and sensor data, and the blockchain storage format is single, making it difficult to effectively process these data; furthermore, historical data management requires long-term data preservation for analysis and backtracking, but storing all historical data on the chain will increase the burden and affect performance; in addition, legal compliance requirements are complex, and the immutability and openness of the blockchain may conflict with data privacy protection and deletion requirements; finally, cross-system data sharing and interoperability require data consistency and traceability, but it is difficult to achieve efficient cross-system data sharing and verification by relying solely on on-chain storage. Therefore, the on-chain and off-chain collaborative storage systems came into being, which store key transaction records and verification information on the chain to ensure data security and integrity and store large-scale non-critical data on the chain to reduce costs and improve efficiency, thereby comprehensively utilizing the advantages of cloud storage and blockchain to achieve efficient, flexible, and secure spatio-temporal big data management and application.  
Blockchain's on-chain and off-chain cooperative storage combine both on-chain and off-chain data storage. Blockchain's on-chain data storage cost is relatively high, and by storing part of the data off-chain, it can significantly reduce the transaction cost and storage cost. Storing data off-chain means that most data is stored outside the blockchain, while only necessary metadata and hash values are stored on-chain for verifying the integrity of off-chain data. This approach reduces the volume of data on the blockchain, lowering transaction costs and storage requirements while maintaining the security and reliability of on-chain data. Off-chain storage typically uses traditional databases or distributed storage systems, while on-chain data is secured using blockchain technology to ensure data immutability and transparency. However, the synchronization of on-chain and off-chain data may lead to data inconsistency problems, and additional mechanisms are needed to guarantee data integrity and consistency. Existing on-chain-off-chain collaborative storage schemes require preprocessing of the stored data during the setup phase, and the length of the generated public parameters is linearly related to the length of the submitted data, which not only increases the computational burden, but also leads to a significant limitation of the transmission efficiency when the data volume is too large, affecting the communication efficiency of the system.  
To address the aforementioned issues, this paper proposes a spatio-temporal big data collaborative storage mechanism based on incremental aggregation subvector commitment in on-chain and off-chain systems, effectively ensuring the consistency and integrity of data stored on-chain and off-chain in blockchain systems.