World number one Osaka beat Belarus' Aliaksandra Sasnovich 6-2 6-3 to reach the Madrid last eight for the first time after number three Halep saw off Viktoria Kuzmova 6-0 6-0 in 45 minutes.

World number two Kvitova defeated Frenchwoman Caroline Garcia 6-3 6-3.

The two-time Wimbledon champion now faces seventh seed Kiki Bertens.

Australian Open and US Open champion Osaka will have to overcome Swiss player Belinda Bencic while Halep will play Australian ninth seed Ashleigh Barty.

Halep, who beat British number one Johanna Konta in the second round, is hoping to win the title for a third time after victories in 2016 and 2017.

"I don't think it was her [Kuzmova's] best match. Maybe she was nervous before the match a little bit," said the Romanian.

"It's a big tournament and it's not easy to face the top players. I've been in that position. I played my best tennis and I felt good on court."

World number eight Sloane Stephens is also through in Madrid after beating China's Saisai Zheng.

The American came from a set down to win 3-6 6-3 6-2 and set up a quarter-final against Croatia's Petra Martic.