Hi,
It would be good to have daily quests in Quick Battle with such goals that are as Objectives in Story mode.
Something like :
And the reward will depend on the difficulty of the quest of some Runes , 5 or 10 crystals for harder sounds like a good scaling 
I just wish QB was easier to understand, when will I get my rewards if any? Do I NEED to fill up the victory/defeat bar(whichever is first?) I’ve seen so many damn rounds end/begin, what is the damn point of qb other than 300 points and whatever the hell those ribbons are for? I haven’t seen anything that resembles a reward or anything other than the 8 hour boosters.
To be honest how quick battle works is clear as day.
Each time you win you get ribbons.
Ribbons let you climb the leaderboard.
When the current QB ends(you can see the “end in” timer on top of quick battle) you get the reward based on your position in the leaderboard.
Rewards can be checked in quick battle easily too.
As for OP: yes a daily quest system would give more reasons to play.
Right now I play I. The last day of QB if I feel like grinding for the first position.
Else I do not care at all beside occasional story to get some runes left(mostly the 5 colors ones)