The Bank for International Settlements summarized several criticisms of bitcoin in Chapter V of their 2018 annual report. The criticisms include the lack of stability in bitcoin's price, the high energy consumption, high and variable transactions costs, the poor security and fraud at cryptocurrency exchanges, vulnerability to debasement (from forking), and the influence of miners.
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The incremental complexity and technological know-how needed for this method are both downsides to the paper wallet approach. Cold storage solutions and hardware wallets are less nimble than other options, too; if the price of bitcoin were crashing, for example, you might find yourself slower to the draw than if you merely kept your BTC on a site like Coinbase.
Generally speaking, every bitcoin miner has a copy of the entire block chain on her computer. If she shuts her computer down and stops mining for a while, when she starts back up, her machine will send a message to other miners requesting the blocks that were created in her absence. No one person or computer has responsibility for these block chain updates; no miner has special status. The updates, like the authentication of new blocks, are provided by the network of bitcoin miners at large.