The structure of protonated acetylene C2H3+ was investigated through a series of ab initio simulations that treat all nuclei as either classical or quantum particles. This makes it possible to probe separately the effects of quantum and thermal fluctuations on the molecular structure of this floppy molecule. The simulations show that the quantum ground state of C2H3+ is best pictured as a quasi-planar bridged structure with anisotropic delocalization of the protons due to zero-point quantum effects. Comparison with high-temperature simulations suggests that recent Coulomb explosion imaging data that led to a different conclusion did not probe the ground state of the fluxional molecule C2H3+.