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In the context of blockchain size manageability, I have seen both the terms sharding and pruning mentioned. Sharding mostly for Monero, and pruning mostly for Aeon. What is the exact difference between the two?

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Pruning the blockchain means removing some data from it, like old transaction signatures and range proofs. For example, your Monero daemon could store an incomplete version of all the blocks older than one week (no signatures, no range proofs) to save some space, and it would still have all the info (inputs, outputs) allowing a wallet to check its balance.

Sharding means means separating the blockchain data in several sets, and storing only one of these sets. For example, if there were a set for blocks with an even block height and a set for odd block height, your Monero daemon could store the data for blocks with an even block height only. In this case, a wallet would have to connect to several nodes (at least one in each set) to get all the info allowing it to check its balance.

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