NEW YORK—Sometimes the best way to understand what separates the world's No. 1 player from the competition is to see her go up against an opponent who can do many of the same things she does.

Madison Keys can do many of the same things that Serena Williams does. She can hit aces—Keys had eight in their fourth-round match, Serena had six. She can hit winners—Keys had 24 to Serena’s 18. And like Serena, Keys can belt the ball past her opponent from anywhere, even when she’s on the run. On at least three occasions on Sunday, all Serena could do as she watched a frozen rope whizz past her was nod her head and say, “Nice shot.”

Yet when it came to the score, Serena ended up well ahead; Keys managed to win just three games in each set. That’s because Serena did all of the other, duller things you need to do on a tennis court—keep the ball in, make your second serve, construct points, move your feet, defend—so much better. Anyone who thinks that Serena is all power, all the time should find a tape of this match. While Keys went for broke whenever she had a chance, and sometimes when she didn’t, Serena kept things in lockdown mode all afternoon. She made just six unforced errors (to Keys’ 19), committed no double faults, faced no break points, and was sparing with her screams and fist-pumps. Even Serena’s post-match twirl for the crowd was all-business.

Afterward, Serena said she knew she couldn’t afford a third straight slow start here. She remembered the early lead that Keys had built when they played in Australia in January.

“It was important,” Serena said of how she came out of the gate in this match, “because she’s such a good player, such a powerful player. She can get off to a fast start.”

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Serena also knew she couldn’t afford a third straight shaky day on her serve—she had 17 double faults in her last two matches combined.

“'It’s now or never, Serena,'” she said she told herself before this one, “you’ve got to get that serve together.”

Serena spent much of the morning patiently practicing that serve with her coach, Patrick Mouratoglou, and it made the difference in the first set. Serena began by holding with an ace and striding regally toward the sideline—her “ace walk,” as it's known. For the next six games, each woman used her serve to dominate the other and hold quickly; after just 19 minutes, the score was 4-3, and a taut, tight affair seemed to be in order.

Just like that, though, the set was blown open when Keys’ serve failed her: At 3-4, she double-faulted twice and was broken. The misses, according to Keys, didn’t come out of nowhere.

“She put a lot of pressure on me to have to kind of almost overplay,” Keys said. “In a sense I feel like she made very little unforced errors, making me hit a lot of winners. I felt like I was going for it more than I normally would have. I felt like if I didn’t hit a winner, I wouldn’t get to the next ball.”

That was never more true than in the match’s other crucial game, which came with Keys serving at 3-3 in the second set. She began it by hitting three straight winners and going up 40-15. One of those winners, a rifle-shot backhand down the line, was the freest shot she had hit all day, and I half-imagined that it might signal the start of a strong run of play from her. Instead, Keys headed in the other direction entirely.

At 40-15, she went for too much on her second serve and double faulted, then followed it up with three straight errors and was broken. This would be the most Keysian game of the match, and the one that doomed her. (And yes, I’m trademarking the phrase “We’re All Keysians Now” as a headline when she wins this tournament in a few years.)

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While Keys tried for winners while backing up and running wide, Serena didn’t feel the need to go for broke. Instead, she hit with pace, depth, and margin; they were rally balls, but rally balls with a sting and a purpose.

“Today she really pushed me,” Keys said, “I didn’t really have the chances I normally have to kind of dictate the point.”

Once again, as she did two years ago against Sloane Stephens in this round, Serena has pushed one of the putative heirs to her American-tennis throne back a step. Not only was Keys unable to beat her, she couldn’t match Serena’s signature trick from last round, the forehand winner she hit while simultaneously doing a full split against Bethanie Mattek-Sands. Today Keys did the split, but missed the forehand.

Even at 34, Serena was the most active player on the court, the one taking the frenetic little steps needed to get in position. On one important point early in the second set, Keys had a rare chance at a short ball, but she was a step slow getting to it. Forced to take it from below the level of the net, she flipped the ball up and over the baseline. Afterward, she turned to her player box, smiled, and stuck out her tongue—she knew what she did wrong. As much as anything else, that one little step forward could be the key to Keys’ future development. A little more footwork would take her a long way

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While Keys goes back to the practice court, Serena moves on, to another quarterfinal and another match with her sister Venus, their 27th. This one is even more fraught than usual. What if Serena’s Grand Slam quest is ended by her sister? What if Venus, after so many years in her shadow, steps out of it now? Venus, after surviving three-setters in her first two matches, has been solid in her last two, one of which was a win over the young star of the summer, Belinda Bencic. Venus also beat Serena as recently as last summer.

But I don’t think she’ll beat her here. That’s not because Venus won’t want to stand in Serena’s way. It’s because in matches like the one she played on Sunday against Keys, Serena can make herself too solid for any opponent. When Serena knows she must be ready right away, when she knows she must hit with consistency as well as power, when she knows she has to be at her best, she typically controls the proceedings. Serena played that type of match in her 6-4, 6-3 win over Venus at Wimbledon earlier this summer. There was very little, I felt, that Venus could have done to alter the result or make the match any closer than it was.

“The only player in the draw I don’t want to play,” Serena said of Venus today, “not only because she’s my sister, but for me she’s the best player."

It’s been said that the highest compliment in women’s tennis is to have Serena bring her A-game against you. I don't think she'll insult her sister be doing anything less on Tuesday.