How quickly could the quarantine on cuban have turned into wwiii?

 If the Soviet Union had elected to run the quarantine/blockade, the US would have fired on Soviet ships. This would have been an act of war, and the USSR would have had no choice but to respond. This would have dragged in both sides' allies, triggering a new world war. And an escalation to a nuclear exchange would almost certainly have occurred, as whichever side was losing in a conventional war would have had little choice but use nukes or surrender.