But without that quantum computers the mining will be too slow to carry-out the heavy load of transactions and mining activities - as a result, single transaction will take forever to complete since there will be a pile-up of data on the blockchain.
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i'm not quite sure i get you..
anyway, data theoretically shouldn't pile up on the blockchain, because the difficulty algorithm should adjust to make mining easier if it starts to head that way.. but like i said that only happens every 2016 blocks.