Your question is sort of backwards. I think you meant to ask, "is there room for a cryptocurrency-run state?" I think that's very easy to answer if you just imagine a government and replace "constitution" with "protocol" in your head.
There is a lot in the constitution that is open to interpretation. This is especially true and society and technology evolves in ways that the constitutional writers could not have imagined.
Bitcoin protocol on the other hand is very black and white. It is very clear if a TX or an output is valid or not.
This is true, and why a change to the constitution would require a real "amendment" (fork). A special "coin" could exist specifically for voting on whether or not the majority approves of a fork, though Proof-of-Whatever providers and users of the actual national coin client could override the decision simply by running the modified solution (or sticking with the old), I suppose, though you could also force auto-updating, turning it from opt-in to opt-out (and there are certainly other ways for a government to force users to use only one version).
Theoretically, too - keep in mind law enforcement could be handled by drones controlled by the protocol with rules "voted on" by people.
This could make revolution fascinating, where it could be a bloodless 100% money/R&D/PR/manufacturing-based war to fork the coin, or a grassroots effort to change the national coin used, which can also result in a total "reset" of the economy for better or worse. Of course, that kind of revolution isn't necessarily bloodless, though a vote to go to war against against rebels would probably require a fork itself to change the rules combat drones operate on, should it be handled by the coin's network, which I can't imagine why isn't possible (combat drones would operate on the instructions pushed in the most recently received block with, say, 50 confirmations).
-Not to say it's a superior solution, but I'll be surprised and disappointed if a country somewhere doesn't implement a similar experimental system before I die. Err - well, I guess I won't be; I'll be dead.
On "government's" side, since this can eliminate the need for dedicated statesmen and even bureaucrats to some degree (replaced largely by developers), this could result in a total intrusion of privacy, but violated by robots with no humans watching (unless a judge requires it), which may be considered acceptable. Assuming tech continues to decrease in cost, it could result in all sorts of absurd situations. For instance, say distilling alcohol in your country is illegal but not considered "actionable against the person" (rather, drones aren't going to ensure you get to a court at gunpoint or issue a fine) -- maybe instead, when the ever-seeing drone network determines you've violated the anti-distilling law, they simply turn your burners/heaters off, or if they're unable to interface with the device, issue a command to your energy provider to cut power if the law isn't complied with, and they could perhaps go from there if the person is still non-compliant (fly the pots away, hover around the offender and issue warnings, etc. There're all sorts of errors which'd be made, so humans would definitely still need to be involved, but the absurd science fiction will basically write itself, maybe with a program called "Bots" ("Cops") which has live feeds of crimes in progress. That could be a punishment... threaten to stream the crime, which could also make up plenty of pornographic content to stream and fund the government... maybe in a theocracy, they'll outlaw sodomy (another "not actionable against the person" crime) but permit streaming of sodomy within and outside the country to subscribers. With eyes everywhere, you could probably fill every porn niche, and the false positives should be equally fascinating.
I'm getting off-topic from the thread-jacking, though -- just one of many examples on "creative funding" for our drone overlords....
Getting back off-topic - assuming AI is capable of following basic real-world "procedural generation," it should be possible for the constitution to define variables with ranges on how the AI government is to act and react with regards to both net revenue and culture. Assuming self-replication is feasible, it can build up, maintain, upgrade, and scrap itself and its units as stated. I suppose it could also be future-proofed, changing some laws real-time based on what it perceives. For instance, maybe a populist government would want certain "non-person-actionable" crimes decriminalized if it logs some percentage of the population committing the crimes. -So say marijuana begins outlawed with a flag where the software requires data be logged and analyzed on the crime, and if, say 15% of currently-living citizens have attempted or successfully committed this crime, the AI itself could repeal that law simply by monitoring cultural shifts.
I suppose it'd be stupid not to utilize widespread drone use's ability to maintain an internet with mesh networking (another revenue source!) - so if a law like this were changed, an update and alert (similar to Bitcoin's emergency alert system, I suppose) could be pushed to users in real-time with the changelog notifying everyone that the law was repealed virtually as soon as it happens, and this could happen without a fork. ... I've forgotten what I was rambling about now, though.