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This question already has an answer here:

Assuming quantum computers become mature enough to effectively crack elliptic curve cryptography of the type used by bitcoin, is Mastercoin secure?

marked as duplicate by David Perry Nov 26 '13 at 18:46

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    This is not a duplicate, see Assaf's own answer as to why this may be different than Bitcoin. – ripper234 Nov 26 '13 at 22:44
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In principle the Mastercoin protocol should be exactly as safe as bitcoin (which it uses as a back end) against the powers of quantum computation. However, there is one important difference. bitcoin addresses that were used at least once are not safe against quantum computers (look for example in this article by Vitalik Buterin). Since all Mastercoins were created by sending bitcoins to the exodus address it follows that any original bitcoin/mastercoin address (by that I mean, the original address from which bitcoins were sent to the exodus address) is not quantum safe.

Should we worry about it? probably not. Quantum computer attacks against bitcoins are not expected in the near future and even if they were here tomorrow it is more likely that attackers use them against bitcoin directly. Hopefully by the time this issue becomes relevant we all have nice mastercoin clients we can use to easily transfer out funds to a never-used-before address which is quantum safe.

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Quantum is currently not mature as it requires helium to follow up so should be safe at this moment.

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