What is the concrete communication complexity of Commitment schemes?
The simplest commitment scheme is a hash based commitment; to commit to a value $M$, the committer selects a fixed length value $R$ and publishes $\operatorname{Hash}(R \mathbin\Vert M)$; to decommit, he publishes the values $R$ and $M$.

If the Hash function is collision resistant, then the committer cannot commit to two different messages; if $R$ is long enough, then it is infeasible to recover $M$ from the commitment.

With this in mind, and assuming we're using a 256 bit hash (such as SHA256) as our hash function, and a 128 bit $R$:


Data sent by verifier: none
The commitment: 256 bits
The decommitment: 128+n bits


This is a total of 384+n bits.

I believe that it can be shown that you cannot do better without reducing the practical security of the system below the "128 bit" level