Kvitova, a former Wimbledon champion and ranked No. 8 in the world, beat Mandy Minella of Luxembourg 7-5, 6-2 to advance to a Friday match against Croat Petra Martic.
Vinci rallied to eliminate Dutchwoman Kiki Bertens 2-6, 6-3, 6-4, and will meet Czech wild card Karolina Pliskova Friday.
Peer will meet Cadantu and German Annika Beck takes on Italian qualifier Maria Elena Camerin in the quarterfinals Friday.
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