In a sharded state (the hypercube one for example), does each shard produce their own blocks (is each shard their own unique block-chain) ?

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Hypercube is no longer planned to be implemented. But yes, each shard produces their own blocks, although the latest plans is to call them collations, which are planned to occur more frequently than blocks. For details see https://github.com/ethereum/wiki/wiki/Sharding-introduction-R&D-compendium and https://ethresear.ch/t/minimal-sharding-protocol-extension/1823.

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