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Why does Ethereum blockchain need computers to agree on blocks? If by means of cryptography, only the private key holder can generate the appropriate hash for a certain transaction, why not let a single miner just include the transaction in the block. Why the need for multiple computers to validate the transaction?

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Giving this a shot but there are probably better answers.

Having more than one miner has advantages of decentralization and no single-point-of-failure. If a miner has an issue, other miners should be able to continue to process transactions. Decentralization also improves censorship resistance: a single miner would be able to censor transactions as they wish.

Multiple computers need to validate the transaction so that the miner cannot include an invalid transaction. Otherwise a miner could create invalid state transitions, like increasing the money they have.

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Your question touches the "why" of Distributed Ledgers vs. Distributed Databases. You fundamentaly use a DL because you operate a Distributed Database in an adversarial environment, without a central coordinator. Consensus is not only for Validation of Transactions but for example also about consensus obout the order of Transactions which manipulate the state of the DL.

Rauchs et al. has a nice Definition of what a DLT is/does:

A DLT system is a system of electronic records that enables a network of independent participants to establish a consensus around the authoritative ordering of cryptographically-validated (‘signed’) transactions. These records are made persistent by replicating the data across multiple nodes and tamper-evident by linking them by cryptographic hashes. The shared result of the reconciliation/consensus process (the ledger) serves as the authoritative version for these records.

And Richard Brown has a nice Article (where he mostly talks about Corda) where he shows the difference between Distributed Databases and Distributed Ledgers. enter image description here

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