How are quad circuit breakers used?

Answer 1

This would typically be seen in older residential installations where the service was of such a high amperage that the mains required larger breakers than were available (for that panel). Placing two breakers in parallel allowing sharing of the load. Two breakers per (split) phase would total four breakers.

Two breakers of 50 amp rating each (in parallel) provide (generally) 100 amps of protection. Two sets of two would provide mains protection for a 200 amp service.