Point Format finalist Kumar defeated Nalin Patel of England 1000-490 while Shandilya got the better off compatriot Sourav Kothari 1000-989 in a close finish.
From the traditional time format, this year championship saw the introduction of 'Long up' format, with 400 points for the league stages, 500 for round of 32, 750 for last 16 and 1000 for quarters.
Kumar will take on Peter Gilchrist of Singapore while former World champion Shandilya will face England's David Causier later in the day.
The results (with high breaks):
Quarter-finals (1000-up): David Causier (ENG) bt Balachandra Bhaskar 1000 (200, 118, 102)-636 (209, 185, 119); Alok Kumar bt Nalin Patel (ENG) 1000 (112, 140, 166)-490; Peter Gilchrist (SING) bt Rupesh Shah 1000 (108, 102, 102)-853 (141, 102, 102); Ashok Shandilya bt Sourav Kothari 1000 (135, 127, 109)-989 (243).
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