In the age of big data, key-value data updated by intensive write streams is increasingly common, e.g., in social event streams. To serve such data in a cost-effective manner, a popular new paradigm is to outsource it to the cloud and store it in a scalable key-value store while serving a large user base. Due to the limited trust in third-party cloud infrastructures, data owners have to sign the data stream so that the data users can verify the authenticity of query results from the cloud. In this paper, we address the problem of verifiable freshness for multi-version key-value data. We propose a memory-resident digest structure that utilizes limited memory effectively and can have efficient verification performance. The proposed structure is named IncBM-Tree because it can INCrementally build a Bloom filter-embedded Merkle Tree. We have demonstrated the superior performance of verification under small memory footprints for signing, which is typical in an outsourcing scenario where data owners and users have limited resources.