  Zero-knowledge proofs have emerged as a powerful tool for enhancing privacy
and security in blockchain applications. However, the efficiency and
scalability of proof systems remain a significant challenge, particularly in
the context of Merkle tree inclusion proofs. Traditional proof aggregation
techniques based on AND logic suffer from high verification complexity and data
communication overhead, limiting their practicality for large-scale
applications. In this paper, we propose a novel proof aggregation approach
based on OR logic, which enables the generation of compact and universally
verifiable proofs for Merkle tree inclusion. By aggregating proofs using OR
logic, we achieve a proof size that is independent of the number of leaves in
the tree, and verification can be performed using any single valid leaf hash.
This represents a significant improvement over AND aggregation, which requires
the verifier to process all leaf hashes. We formally define the OR aggregation
logic, describe the process of generating universal proofs, and provide a
comparative analysis demonstrating the advantages of our approach in terms of
proof size, verification data, and universality. Furthermore, we discuss the
potential of combining OR and AND aggregation logics to create complex
acceptance functions, enabling the development of expressive and efficient
proof systems for various blockchain applications. The proposed techniques have
the potential to significantly enhance the scalability, efficiency, and
flexibility of zero-knowledge proof systems, paving the way for more practical
and adaptive solutions in the blockchain ecosystem.
