It won’t save drafts. More specifically, it won’t save drafts from the dashboard, though you can publish them, and you can edited previously save drafts.
If you save a draft from the dashboard, it creates what appears to be a copy of the last draft you have before than. Clicking on that draft brings up that older draft, and the duplicate then disappears from the list of drafts when you go back.
This is on WordPress 3.7.2, which was working yesterday.
I’m hoping this is a transient error…
Update: Upgraded to 3.8.2, and Quick Draft still appears to be broken…
it seems to work for me, but I’m running 3.8.2.
It’s broken in both 3.7.2 and 3.8.2. It’ll be fixed in 3.9, which will be released next week. We may release 3.8.3 in the interim to fix this, as losing any of your content is a serious bug.
(Also, sorry for all of the version numbers. To explain, 3.7, 3.8, and 3.9 are major releases which came out in October, December, and next week, respectively. 3.8.2 was a security release from earlier this week. At the same time we also released an update for sites still running 3.7.x, which was 3.7.2. So you’ll be jumping three major versions in just two weeks. Enjoy!)