Alternatives to PoW are centralization (i.e. Linden Dollars or national currencies) and proof-of-stake (Ethereum). With centralization, there is the problem of

Alternatives to PoW are centralization (i.e. Linden Dollars or national currencies) and proof-of-stake (Ethereum). With centralization, there is the problem of trust. The issuer can just dilute the currency supply, go bankrupt, get bribed, or shut down. With proof-of-stake, the tokens themselves are used to distribute trust. But during an interruption of service or a network split, if the two subnets diverge, there is no objective way to decide the most legitimate state of the tokens. Choosing the wrong one might end in losing your stake. It might keep working with long-term trusted peers and reliable networks. But if those peers become centralized enough, they can then be bribed. Proof-of-work provides an objective and unforgeable measure of legitimacy: work i.e. resources burned. In absence of a legitimacy measure, the network is vulnerable to Sybil attacks.