  Quantum superposition of high-dimensional states enables both computational
speed-up and security in cryptographic protocols. However, the exponential
complexity of tomographic processes makes certification of these properties a
challenging task. In this work, we experimentally certify coherence witnesses
tailored for quantum systems of increasing dimension, using pairwise overlap
measurements enabled by a six-mode universal photonic processor fabricated with
a femtosecond laser writing technology. In particular, we show the
effectiveness of the proposed coherence and dimension witnesses for qudits of
dimensions up to 5. We also demonstrate advantage in a quantum interrogation
task, and show it is fueled by quantum contextuality. Our experimental results
testify to the efficiency of this novel approach for the certification of
quantum properties in programmable integrated photonic platforms
