Vector Commitments are powerful primitives that find applications in:

• Proof of (Persistent) Space (PoS): a distributed storage network based on a blockchain mechanism, where users can provide storage capacity for the network, and thereby earn rewards by periodically producing cryptographic proofs that certify that they are honest.

• Stateless Cryptocurrency: a system where neither validators of transactions, nor cryptocurrency users do not need to store the full ledger state

• Verifiable Small Computation on Big Data: a system that allows to publicly and efficiently verify the correctness of low-complexity queries on a large dataset stored by an untrusted third-party.

Known constructions of VC have proof sizes either constant or logarithmic in the size of the vector. They rely on various assumptions, such as bilinear groups, RSA or class groups, hash functions and lattices.
Some important properties for VC schemes are aggregation of different opening proofs, updatability of commitments and proofs, homomorphism of opening proofs, transparent setup for generating public parameters (as opposed to trusted setup).

Ideally, we would like to construct VC schemes that have:

• minimal public parameters size, with minimal trust requirements on their generation

• efficient proof generation (or time-space trade-offs) and efficient verification

• proof-size independent of both the vector’s length and the number of opened positions.