Kuchar played bogey-free in a round of seven-under 63. He lost a chance to add one more birdie when he drove into a fairway bunker and narrowly cleared the lip.
Simpson reached the par-five 18th green in two for a two-putt birdie with his conventional putter. That gave him a 66 and a share of the lead at 12-under 128.
Tim Clark (65) and Troy Merritt (64) played in the morning and were at 10 under. Defending champion Jimmy Walker had his 15th straight round in the 60s in Hawaii. He had a 66 and was four shots behind.
Rory McIlroy made his first competitive hole-in-one as a professional and shot a six-under 66 to move into third place in the Abu Dhabi Championship in the United Arab Emirates, two strokes behind second-round leader Martin Kaymer.
The top-ranked McIlroy threw his hands in the air and high-fived playing partner Rickie Fowler after his nine-iron from 177 yards on the 15th hole landed a yard from the hole, hopped right and trickled in.
Kaymer the U.S. Open champion who won the Abu Dhabi event in 2008, 2010 and 2011, had a 67 to reach 13 under. Belgium's Thomas Pieters was second after a 67. Fowler was tied for 56th at two under after a 75.