The relative influence of thermal and quantum fluctuations on the proton transfer properties of the charged water complexes H₅O₂⁺ and H₃O₂⁻ was investigated with the use of ab initio techniques. These small systems can be considered as prototypical representatives of strong and intermediate-strength hydrogen bonds. The shared proton in the strongly hydrogen bonded H₅O₂⁺ behaved in an essentially classical manner, whereas in the H₃O₂⁻ low-barrier hydrogen bond, quantum zero-point motion played a crucial role even at room temperature. This behavior can be traced back to a small difference in the oxygen-oxygen separation and hence to the strength of the hydrogen bond.