He has won several chess competitions. In his early years, Krivoshey won the 1998 Ukrainian Under-20 Championship. In 2007, he outperformed his peers and won in the Cappelle-la-Grande Open, a competition in France.
In 2008, Krivoshey participated in the Corus tournament. He scored 7.5/13 and took the second place on countback to Jan Smeets. His most recent victory was in 2017, in the St.Petersburg Summer Open.