What, exactly, does a blockchain have to do with this? ISTM all that's needed is a spreadsheet with an audit log, and even the audit log is somewhat optional.
Using a blockchain for the log would work, but so would any other means of checkpointing the thing periodically and verifying that the audit log is really append-only.
IMO the real benefit of a blockchain is that an owner of some interest in a company could plausibly validate that their interest is actually logged in the source data for the cap table. But this would depend on having some way for the owner to know the current blockchain head, and this isn't going to happen by magic.
Blockchains, in theory, could also serve as a voting mechanism on splits, which could be manifested as blockchain hardforks. As Satoshi Nakamoto wrote:
> The proof-of-work also solves the problem of determining representation in majority decision making. If the majority were based on one-IP-address-one-vote, it could be subverted by anyone able to allocate many IPs. Proof-of-work is essentially one-CPU-one-vote. The majority decision is represented by the longest chain, which has the greatest proof-of-work effort invested in it.
Essentially a blockchain can codify the ownership of a stock share into a smart contract.
That's the theory, and only insofar as it applies to a Proof of Work blockchain like Bitcoin.
A private blockchain is more likely to be a Proof of Stake blockchain, in which shares (or tokens of other sorts) are issued in a "premine" which creates all of the tokens that will ever exist. These are then distributed to the initial group of trusted members of the private blockchain. From there, the blockchain can eliminate the need for a trusted authority to serve as the system of record of ownership.
Google Docs has a change log could be a first step.