The Mohs hardness scale is the standard geologists and gemologists use to grade minerals and gemstones. They use the Mohs hardness test to determine "scratchability" of a material. So because diamond can scratch quartz, diamond is considered harder than quartz.
It's pretty easy to perform this test in a lab or on your kitchen table. Start with two specimens with unmarred surfaces. Hold one motionless against the table surface. Drag a sharp point of the second specimen firmly across the surface of the other. If a scratch appears, the second specimen is harder than the one on the tabletop. If not, the second specimen is either softer or the same level of hardness. In that case, reverse the test to see whether specimen one can scratch specimen two.
Just make sure what you are calling a scratch is a groove in the specimen and not residue of a soft material used to do the test.