4 quadrature clock vs 2 quadrature clock + falling edges
"The thing that's been puzzling me with multiphase clocks is : why using a four-phase clock in quadrature {0°, 90°, 180°, 270°} when you can just trigger your process on the falling edges of a two-phase clock in quadrature {0°, 90°}?"

One reason is that in sampling applications, the edges used for sampling need to be precisely, and evenly spaced (in time) from one another.  For example, if you want to sample something at 400 MHz using 4 phases of a 100 MHz clock, you want those rising edges (assuming that's what's being used for sampling) to be at T0, T0+2.5000ns, T0+5.0000ns, and T0+7.5000ns (may have left off some zeros here).  If that is not the case, then you introduce phase noise into the sampled quantity which affects post processing and analysis.

If you try to do that using both edges of a 200 MHz (in this example) clock, the duty cycle of the clock needs to be precisely controlled to be 50.000%.  This is much more difficult to do than generating 4 phases of a 100 MHz clock offset by 90 deg (2.5 ns) from each other.