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I need a cryptographic hash of the entire Ethereum database (of the information in all blocks up to a certain block number). Is there a service that calculates this hash?
It depends what you mean by that.
The block hash is the hash of ordered blocks and their ordered transactions up to the block height.
If you want to easily prove something outside what is already provided then you might have to consume the chain and organize appropriately.
Hope it helps.
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