I am exploring this Bitcoin thing and yesterday installed the Bitcoin-Qt. The first thing that I noted, is that the after syncing with the network that took me 12 hours, the database grew to 13GB. I know that these are all the blocks in a chain from the beginning of time. My question is what is the sense and the theoretical practicality behind this approach? Imagine if all the world will start to use bitcoins, the blockchain will grow to terabytes and will multiply exponentially on a daily basis. Imagine if USA will embrace Bitcoin as its official currency, the size of blockchain will grow few Petabytes daily. The bottom line is that at these sizes it will become impossible to validate transactions and even simply store or use the blockchain.

My question is if I am missing something here, because I don't see any long term and general usage capabilities here?

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    IMHO, this is one of about 2-3 flaws with BTC. One big selling point for bitcoin is that it's "decentralized". If the popularity of bitcoin continues, "super-nodes" will need to be implemented and people will begin to prefer online, centralized bitcoin generation mechanisms, payment processors and wallets. The "dream" of cryptographic currency, for many of us, is to have a currency that is as completely decentralized as is possible. Bitcoin, as a tech, was a wonderful start. However, I do not foresee bitcoin being the tech for such a long term goal.
    – RLH
    Nov 4 '13 at 20:04

Work is being done on cleaning up the database by getting rid of obsolete data, although full nodes would probably always want to keep everything.

Also note that non-full nodes, like a wallet app on your phone, don't need the full 19GB at all and are really quick already.

Also keep in mind that disk size and network speed keep growing as well, so we still have quite some time to go before this would really become a problem.

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