In theory, an open blockchain like Bitcoin eliminates the need for a trusted authority to be the system of record. A spreadsheet needs someone to manage it. That person, and that data, must be very trustworthy, else they can simply change the data for their benefit. The compliance procedures that assure trustworthiness cost money. Lots of money. And they still fail.
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.