Upon arriving in Canada, he earned a National Master ranking within a few months at age 11, making him the youngest Canadian to achieve this level.

He was training during this time with Yan Teplitsky, who had studied in the famed Russian school run by Mark Dvoretsky before moving to Canada.

Bluvshtein's first major Canadian success came in 2000, when he tied for 2nd-3rd places in the Toronto Closed Championship, with a score of 8/11 points, behind Eduardo Teodoro IV.

Staying on for the Canadian Open Chess Championship, also at Sackville, Bluvshtein tied for 3rd-7th places, with 7½/10, behind only winners Tony Miles (in his last tournament before his death a few weeks later) and Larry Christiansen.

Then, in the Canadian Open Championship, Montreal 2002, he tied for 4th-10th places, with 7½/10, behind only winners Jean-Marc Degraeve, Pascal Charbonneau, and Jean Hébert.

Bluvshtein's first grandmaster round-robin tournament was the 2002 Montreal International, where he tied for 10th-11th places scoring 4/11; the winner was Degraeve.

Just a couple of weeks later, in the 2nd Chess'n Math Association Futurity in Toronto, he tied for 1st-4th places, with 6/9, along with Yuri Shulman, Walter Arencibia, and Dmitry Tyomkin, missing a norm for the title of Grandmaster by half a point.

In June 2003, Bluvshtein scored his first norm for the title of Grandmaster at a round-robin tournament in Balatonlelle, Hungary, by winning his last three games and finishing with 6½/9.

With funding assistance from chess patron and businessman Sid Belzberg, Bluvshtein was able to work with Israeli Grandmaster Alexander Huzman, and this provided the impetus for his next qualitative advance.

The next month, at the 2004 Montreal International, he made his third and final qualifying grandmaster result with 6½/11 to place fourth; the winner was Zahar Efimenko.

Then he placed third, following a tie-break playoff, at the Zonal Canadian Championship in Toronto, with 6½/9, behind co-winners Charbonneau and Eric Lawson.

He was awarded the title by FIDE at age 16, during the 36th Chess Olympiad, held in Calvià, where he made a further grandmaster norm.