Kvarstad, who competes for Byåsen IL , made his international debut on February 9, 2016 in the Youth Cup competition in his hometown Trondheim . About eleven months later, he started in the Continental Cup for the first time , but missed the points. He achieved this for the first time in January 2018 in Klingenthal when he finished the sprint on the large hill in 27th place. A few weeks later he took part in the 2018 Junior World Championships in Kandersteg . He won the bronze medal in the team competition together with Jens Lurås Oftebro , Andreas Skoglund and Einar Lurås Oftebro . In addition, he was nominated for the sprint in which he was sixteenth. At the end of the season he won the bronze medal in the team sprint at the Norwegian Championships 2018 together with Espen Bjørnstad .
Although Kvarstad was disqualified on his debut in the 2018 Grand Prix , he easily reached the points on the following day with 22nd place. In the 2018/19 Continental Cup season , Kvarstad was a regular part of the Norwegian team and was able to make it into the top ten twice. At the 2019 Junior World Championships in Lahti , he finished seventh in the sprint and 21st in the Gundersen singles. He was not used in the team competition. He finished the season in 29th place in the overall Continental Cup ranking.