Apparently nobody on the first google page has counted Shakespeare quotes. Very well, I could watch new anime, but I also could count Shakespeare quotes in Zetsuen no Tempest. Luckily they are easy to pick out from the regular talk. I watched the episodes in 2x and with skipping, so I could easily missed something. If you remember anything that's missing, please tell me.
Episode 01
01:07 (Hamlet)
Yoshino: The time is out of joint.
Aika: O, curse spite
Yoshino: The time is out of joint. O, curse spite. That ever I was born to set it right!
21:38 (Hamlet)
Yoshino: The time is out of ...
Mahiro: The time is out of joint.
22:08 (Hamlet)
Mahiro: The time is out of joint. O, curse spite. That ever I was born to set it right!
Episode 02
21:31 (Hamlet)
Mahiro: O anger, turn to heat, dry up my brains (original: O, heat, dry up my brains)
This is followed by some other sentences that sound shakespearish, but aren't actually in hat form in any of his works
Episode 03
None found
Episode 04
None found
Episode 05
8:39
This is not a quote, but another important point that returns once in a while is Aika explains the structure of dramas, and more specific tragedies and comedies. The show plays a while with the open thought whether it's story will turn out a tragedy or a comedy.
Aika: Everything happens for a reason. The daily tragedies and misfortunes are all meaningful events, leading towards an ideal conclusion. With his in mind, there probably isn't really any meaningless misfortune.
9:31 (Hamlet)
Aika: There are more things in heaven and earth, Horatio, than are dream of in your philosophy
Episode 06
None, but we have Junichirou holding a skull [link to picture]. Which could be chance, but since the show is so close to Shakespeare I want at least mention it.
That one third of the show for now, I will continue tomorrow, pretty cure is waiting on me. So far no verbal mention of The Tempest, only Hamlet.
Also: can someone who know more about flowers than me identify these flowers?
Well to be fully true to the intent of the show. I think it's supposed to be a clash between what happens in Hamlet and what happens in The Tempest and which result would be most preferred for and by the characters. Certain characters see the future outcomes to end like in Hamlet while others see the futures outcome in reference to what happens in The Tempest and there is a battle so to speak for each future to be the one to come out on top.
The Tempest becomes much more prominent in the 2nd half and would be the better outcome for everyone if it came out on top. The name of the show speaks refers to the future the MC's want.