I could see if bitcoin was the backbone of our financial system (which I believe is starting right now with the NASDAQ trials) the U.S. Government could possibly become the world's largest miner in order to help assure the security of the bitcoin network. They wouldn't sit by idle while China does the majority of the mining. Then I think other governments would join in to insure their own security as well.
This could also provide revenue for the governments possibly alleviating the tax burden placed on citizens.
Is this a possible outcome??
Why should country doing business? That sounds like a socialist country. You think USA will join Cuba and China on their road to Socialism? It proved many times in past private owned companies does much better then country owned, since actual politic elite will always do first for their good then for country good.
Japan, South Korea, and Singapore's governments were heavily involved in business during the industrialization of those countries. I wouldn't call it socialist. It's more like state capitalism as they were using state resources to try to actually make money.