"Proof of Work" used by Bitcoin is a competitive consensus algorithm. Each node races to solve a difficult puzzle first. Doing so earns the right to produce a block and you are rewarded in Bitcoin. The block is where the transaction (value of data) is written and confirmed. However, this race is a waste of time and money for those that don’t win. You get nothing unless you are the first to solve the puzzle. Since no one wants to lose, nodes started working together to solve the puzzle and share the reward based on your computational power (the hash rate).
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Public blockchains are open, and therefore are likely to be used by very many entities and gain some network effects. To give a particular example, consider the case of domain name escrow. Currently, if A wants to sell a domain to B, there is the standard counterparty risk problem that needs to be resolved: if A sends first, B may not send the money, and if B sends first then A might not send the domain. To solve this problem, we have centralized escrow intermediaries, but these charge fees of three to six percent. However, if we have a domain name system on a blockchain, and a currency on the same blockchain, then we can cut costs to near-zero with a smart contract: A can send the domain to a program which immediately sends it to the first person to send the program money, and the program is trusted because it runs on a public blockchain. Note that in order for this to work efficiently, two completely heterogeneous asset classes from completely different industries must be on the same database - not a situation which can easily happen with private ledgers. Another similar example in this category is land registries and title insurance, although it is important to note that another route to interoperability is to have a private chain that the public chain can verify, btcrelay-style, and perform transactions cross-chain.
I have a hard time swallowing that Bitcoin “isn’t a ledger”. That’s like saying “Bitcoin isn’t the blockchain”, and if you take the blockchain away from Bitcoin, you aren’t really left with much (including, sidechains). Perhaps Bitcoin isn’t a ledger *from the perspective* of individual transactions, but by the same logic, nothing that isn’t transaction data is.
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Blockchain-based smart contracts are proposed contracts that could be partially or fully executed or enforced without human interaction. One of the main objectives of a smart contract is automated escrow. An IMF staff discussion reported that smart contracts based on blockchain technology might reduce moral hazards and optimize the use of contracts in general. But "no viable smart contract systems have yet emerged." Due to the lack of widespread use their legal status is unclear.