By a strange twist of fate, three of the four quarterfinal matches are the same matchup: Blue-White Control against Mono-Blue Devotion. Here's a brief rundown of how they went.
Alexander Hayne vs. Eugene Hwang
In the first game, Hwang had Hayne under early pressure, but Hayne managed to keep up with a string of answers. He resolved a small Sphinx's Revelation, but was still down to 7 life facing Thassa and Mutavaults. The Revelation was a good one. Detention Sphere stopped Thassa, and Jace, Architect of Thought took the sting out of the Mutavaults. Next up was Elspeth, Sun's Champion to provide a string of blockers and take over combat. Hwang's last gasp came a turn or two later with an overloaded Cyclonic Rift, but Hayne was ready with Dissolve. His soldier tokens marched to victory.
Hayne 1 - Hwang 0
Hwang resolved a turn-three Nightveil Specter on the play and Hayne didn't have an immediate answer on two lands and a guildgate. The Specter stole a Dissolve from Hayne's deck, but that was no good against Supreme Verdict. Hwang tried to build up a base of devotion in play but each time Hayne would be ready with another Supreme Verdict. Hwang kept the pressure up, but soon Elspeth was hitting the field and Hwang couldn't get past her.
Alexander Hayne defeats Eugene Hwang 2-0
Robert Gillespie vs. Adam Ruprecht
Ruprecht's draw in game one was long on land and short on threats. He could only stare forlornly at his two Bidents of Thassa, one in hand, one in play, and a big stack of island. Gillespie meanwhile was resolving a big Sphinx's Revelation and then Elspeth, Sun's Champion.
Gillespie 1 - Ruprecht 0
In the second game, Judge's Familiar got thassa on board past Essence Scatter, but Ruprecht just couldn't keep any creatures on the board to go with her. Frostburn Weird fell to Last Breath. Syncopate stopped Bident of Thassa. Not even an anemic Master of Waves for two was allowed to stay in play. And then the dynamic duo showed up: Jace, Architect of Thought and Elspeth, Sun's Champion. Ruprect played it out to the bitter end, and then congratulated Gillespie on advancing to the semifinals.
Peter Sundholm vs. Mike Vasovski
Vasovski's Blue-White was a little slow, so as it matched answers to Sundholm's threats, he was also taking a lot of damage. He finally managed to Detention Sphere away Thassa and then Supreme Verdict a turn later. However, he was very low on life, and Sundholm had Mutavaults, stealing the game before a big Sphinx's Revelation could turn the tide.
Sundholm 1 - Vasovski 0
Sundholm mulliganed to six and started slowly with just a third-turn Thassa and then a Frostburn Weird without a fourth land. Vasovski went for Jace, Architect of Thought and plussed it, but Sundholm had Tidebinder Mage in hand. Thassa and the Weird attacked, and with a pump Jace was in the graveyard.
Vasovski played a second Jace and this one served up cards, but crucially, no Supreme Verdict. Sundholm's team dispatched the second Jace and dealt damage to Vasovski. Again no Wrath for Vasovski, but Archangel of Thune. Sundholm swung all out. The Archangel stepped in front of the Weird, and Sundholm fully pumped it to trade. Thassa was turned off, for the time being. A turn later Sundholm dropped Frostburn Weird and hit with Thassa, then lost a counter war over Azorius Charm. Unfortunately for Vasovski, he had no way to stop her hitting play again, and though Elspeth, Sun's Champion could provide board control, Thassa would let Mutavaults slip through for lethal. He soon conceded.