Medvedev earned a 7-6(5), 6-0 win over Germany's Dominik Koepfer while Alcaraz took care of Italian Lorenzo Musetti with a 6-3, 6-3 win in 85 minutes.
Second seed Jannik Sinner of Italy also moved into the last eight beating Christopher O'Connell of Australia 6-4, 6-3.
Alcaraz, who is searching for the 'Sunshine Double' after winning in Indian Wells, was too much for 23rd seed Musetti although the Italian certainly contributed to a crowd-pleasing contest.
Stadium court erupted after a rally in the third game of the second set saw Musetti clip the ball between his legs, lobbing Alcaraz but the Spaniard returned it with a 'tweener' only to lose the point to a deft volley at the net.
But overall it was a comfortable match for Alcaraz whose all-round game looks in strong shape.
"I tried to play my game, to play aggressive shots, go to the net, drop shots, my style in general and I think I did pretty well and I am happy with the performance," said the Spaniard, who was enthusiastic about his form.
"I don't know if this is the best game I've played but without a doubt it is the best feeling. I feel great on the court, I'm moving great, I am not injured and not thinking about my ankle anymore.....it's the best feeling since the summer," he added.
In the quarters, Alcaraz will face Bulgarian Grigor Dimitrov, who beat Pole Hubert Hurkacz, triumphing in a third set tie-break for a 3-6, 6-3, 7-6(3) win.
Slow start
Medvedev made an unusually sloppy start on Stadium court, with a series of unforced errors, and found himself trailing 4-0 in the first set tie-break.
But having overturned that deficit, Medvedev went on to win all the remaining games as he showed that, despite his concerns over the rapid deterioration of balls on the hard court surface, he is comfortable with the conditions.
The Russian suggested that Koepfer had struggled to recover from the blow of seeing his tie-break lead vanish.
"I think sometimes it happens, when you lose the first set the way he lost it. He played very good, probably was closer to winning it because of the 4-0 in the tie-break and when you lose such a set, especially when the points were tough, it brings your energy down," he said.
"I knew I had to use that in the beginning of the second set. That's why it was the most important and I managed to do it. Sometimes that happens, but the level was high today," he added.
The world number four, Medvedev will face Chilean Nicolas Jarry who upset Norwegian seventh seed Ruud with a 7-6(3), 6-3 win.
Cheered on by a large contingent of Chilean supporters, Jarry had 39 winners to Ruud's 17 and was delighted at his progress into the last eight.
"I'm very happy with the way I played. I was very strong on court, not missing many easy balls. The conditions were so tough. The wind was extreme and Casper also, he's a grinder," he said.
Sinner trailed 3-1 in the first set, as the Australian cleverly mixed up his approach, asking a variety of questions of his Italian opponent.
He faced a potential double break at 0-30 but responded strongly to come back and take the set after 58 minutes.
The second set was more straightforward for Sinner who broke O'Connell's first service game and then took care of business to wrap up the win.
"He started off really well, I made a couple of mistakes. When you are a break down especially in the beginning, it's always tough, also mentally," said Sinner.
Sinner will face Czech Tomas Machac who eased into the quarters with a 6-3, 6-3 win over another Italian, Matteo Arnaldi.
