If you start Quest Battles through the main menu, you'll see a list of all available quest battles, and the ones you have done are marked with a green checkmark.
This is not consistent though.
As you can see on this screenshot I've done two of the Orion battles (I did them in a solo Heralds of Ariel campaign, *not* through the Quest Battle) and they get marked as done correctly:
But if you look at Grombrindral, I did all four of this quest battles in a co-op campaign, none are marked as done:
I've also done 3 of Thorgrim's battles in that same co-op campaign, none marked as done:
But on the other hand, in a co-op campaign where my co-op partner was Ogre Kingdoms and I participated in his quest battles, are marked as done:
It seems to be very, very inconsistant. Some are marked as done, even if it's the battle of my co-op partner and not actually mine (I never played Ogre Kingdoms), and others I have done in my own campaign, like Grombrindal, and none are marked a done. It's weird.
Please fix this, I'd love to do every battle at least once through campaigns, but have no way to track (without documenting stuff myself) if the checkmarks are this buggy.