"Dirty" Bitcoins/Find Bitcoin origins?
The tool for complete history of all bitcoin minted already exist and its called blockchain. You can see all history here (random transaction selected)

https://blockchain.info/tx/ff698f3e5321448d4d889fcd3c91f9e5f5767542d2f0fd7e4aa41a83abec3ab7

There is a term taint and it's meaning how two addresses are connected. It partially explained here What are tainted coins exactly?

Bitcoin is not a legal tender and nobody have to accept any bitcoin - with normal dollar any dollar is a dollar that is worth always the same and cannot be legally tainted with crime. It's more legal question that a technical one because government could not adhere to technology reasons.