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Five-times champion Williams lost to Germany's Sabine Lisicki to snap her 34-match winning streak, leaving only one member of the world's top four left in the tournament.
With fourth seed Agnieszka Radwanska in the other half, the 20-year-old Stephens is now being talked about as potential Wimbledon champion a bit sooner than expected.
The only American to reach the last eight of the men's or women's singles refused to be sucked in by the hype, though, after a 4-6 7-5 6-1 win over Puerto Rican Puig with whom she has trained with regularly at the same Florida base.
"I saw (Kirsten) Flipkens, she fell to the ground today. You would have thought she just won Wimbledon," Stephens, who reached the Australian Open semi-final this year having beaten Williams in Melbourne.
"I think I was just kind of the whole match really calm. I was happy to get the win, so I wasn't too like overjoyed."
Seventeenth seed Stephens has a few grand slam titles to win before comparisons with 16-times major champion Williams can be taken seriously but she lacks nothing in confidence.
She even accused Williams of "scaring people" and "intimidation" earlier this year in a Time magazine article.
Asked her reaction on Serena saying in her press conference that she could go on to win the title, Stephens offered an almost sarcastic "Thanks".
"I'm top 20 in the world for a reason," she said when asked about a recent return to form after a dip. "I didn't like all of a sudden snap my fingers and I got good.
"I put in a lot of work, took a lot of sweat, like bad hair days, all that other stuff, to get to where I was. I realise that I just couldn't let that go to waste."
Stephens, the highest-ranked of a growing bunch of American women in the top 100, said thoughts of winning the title had not yet entered her head, especially with former runner-up Marion Bartoli up next and then possibly 2011 champion Petra Kvitova.
"Playing a grand slam, every occasion is big, every quarter is big, even if you're playing like Timbuktu Court, Aorangi," she said.
"I'll go out and play hard. I'm looking forward to it. I'm excited. Hopefully I'll have a good match and have fun.
Results from the seventh day of Wimbledon (x denotes seeded player):Men's singles:
Fourth round:
Novak Djokovic (SRB x1) bt Tommy Haas (GER x13) 6-1, 6-4, 7-6 (7/4)
Tomas Berdych (CZE x7) bt Bernard Tomic (AUS) 7-6 (7/4), 6-7 (5/7), 6-4, 6-4
David Ferrer (ESP x4) bt Ivan Dodig (CRO) 6-7 (3/7), 7-6 (8/6), 6-1, 6-1
Juan Martn Del Potro (ARG x8) bt Andreas Seppi (ITA x23) 6-4, 7-6 (7/2), 6-3
Lukasz Kubot (POL) bt Adrian Mannarino (FRA) 4-6, 6-3, 3-6, 6-3, 6-4
Jerzy Janowicz (POL x24) bt Jurgen Melzer (AUT) 3-6, 7-6 (7/1), 6-4, 4-6, 6-4
Fernando Verdasco (ESP) bt Kenny De Schepper (FRA) 6-4, 6-4, 6-4
Andy Murray (GBR x2) bt Mikhail Youzhny (RUS x20) 6-4, 7-6 (7/5), 6-1
Women's singles:
Fourth round:
Sabine Lisicki (GER x23) bt Serena Williams (USA x1) 6-2, 1-6, 6-4
Kaia Kanepi (EST) bt Laura Robson (GBR) 7-6 (8/6), 7-5
Agnieszka Radwanska (POL x4) bt Tsvetana Pironkova (BUL) 4-6, 6-3, 6-3
Li Na (CHN x6) bt Roberta Vinci (ITA x11) 6-2, 6-0
Sloane Stephens (USA x17) bt Monica Puig (PUR) 4-6, 7-5, 6-1
Marion Bartoli (FRA x15) bt Karin Knapp (ITA) 6-2, 6-3
Petra Kvitova (CZE x8) bt Carla Surez (ESP x19) 7-6 (7/5), 6-3
Kirsten Flipkens (BEL x20) bt Flavia Pennetta (ITA) 7-6 (7/2), 6-3
