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Can I find (always, without fault) the specific set of blocks from which the Bitcoins that I hold originated?
If yes, would such a trace-back be computationally costly?
Tracing Bitcoins to their original blocks
Answers
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Yes you can, every transaction in the bitcoin network is registered in a public ledger called blockchain. You can query this ledger via http interface like https://blockchain.info/api/blockchain_api
I don't think will be costly but it depends how many hands your bitcoin passed. When you find transaction without a sender, you have to stop, this are mined (generated) blocks
3It's not really clear what that means. For example, say a transaction pulls in the 25 Bitcoins mined in two different blocks and, among other outputs, pays out 1 Bitcoin. What does it mean to ask which of those two original blocks is the one that Bitcoin "originated" in? – David Schwartz – 2014-04-13T18:54:27.070