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The Ethereum blockchain is the network’s shared transaction record, and all full nodes and validators keep their own copy of it. This ensures all current peers are always in agreement about who owns what (in other words, the Ethereum state of user accounts and balances). Agreement on which blockchain is the correct one is ensured through Ethreum’s protocol rules.
One of the main protocol rules is that the correct blockchain is the chain whose transactions are all valid and whose blocks have the most accumulated, actively validating 'stake' committed to it. As long as all users have the same blockchain, and are operating under the same protocol, they will all independently arrive at the exact same state of ether ownership (again, the Ethereum state).