Ash Barty and Coco Gauff will face off for the first time in the quarter-finals in Rome, while Iga Swiatek is also into the last eight.
Magda Linette dominated the first set against Petra Kvitova in Rome, but the Czech responded emphatically to reach the second round.
Sofia Kenin, Belinda Bencic and Elise Mertens were all beaten at the Volvo Car Open, where Ash Barty moved through.
Two-time grand slam champion and sixth seed Garbine Muguruza dispatched qualifier Magdalena Frech 6-1 6-3 on Tuesday.
Coco Gauff's Dubai Tennis Championships hopes were ended on Thursday, but Elise Mertens complete a great fightback.
Garbine Muguruza and Coco Gauff progressed to the quarter-finals of the Dubai Tennis Championships on Wednesday.
Coco Gauff annoyed opponent Marketa Vondrousova with remarks to the umpire in their spicy Dubai clash, which the American won in three sets.
Clara Tauson will climb into the WTA top 100 for the first time after the former top junior completed a perfect week in Lyon.
Iga Swiatek reached her first final since winning the French Open and will tackle Belinda Bencic after the Swiss edged out Coco Gauff.
Danielle Collins' injury allowed Iga Swiatek a straightforward progression in Adelaide, but Coco Gauff had to rally to remain alive.