The quad antenna is a development of several inventions.
In 1924, Moses Jacobson patented a loop antennas with rhombic shape.
In 1938, George Brownet al. patented a loop antenna with rhombic shape and quarterwave sides.
In 1951 Clarence C. Moore, W9LZX, a Christian Missionary and engineer at HCJB developed and patented a two-turn loop antenna that he called a "quad". He developed this antenna to resolve issues caused by large coronal discharges while using a beam antenna in the thin air of higher altitudes. Moore describes his antenna as "a pulled-open folded dipole". While the main point of Moore's patent was the two turn single loop design which is not the antenna termed "quad" today, the patent does include a mention and illustration of a two element unidirectional "quad", and describes the time when the full wave loop concept was developed:
In 1957 James Sherriff McCaig patented what we know as a "cubical" multi-band quad antenna.
In 1971 Hans F. Ruckert invented the "Mono-Loop Tri-Band Cubical Quad".
Advantages over a Yagi–Uda
Rigorous testing of the quad antenna show the following advantages over a Yagi–Uda antenna.
Polarization
It is easy to change polarization from vertical to horizontal.
Multiband antenna
It is easier to design a multiband quad antenna than a multiband Yagi antenna.
Higher gain
The 2-element quad has almost the same gain as a 3-element Yagi: about 7.5 dB over a dipole. Likewise, a 3-element quad has more gain than a 3-element Yagi. However, adding quad elements produces diminishing returns. Quoting from William Orr, "Whereas parasitic beams having twenty or thirty parasitic directors are efficient, high gain antennas, it would seem... that maximum practical number of parasitic loop elements for the quad array is limited to four or five."
Radiation resistance is affected by antenna height above ground, element spacing, and environmental conditions. However, values will be higher than for a Yagi and more closely matched to a 50 Ohm coaxial feed.
Lower boom height
"A two-element, three-band quad, with elements mounted only 35 feet above ground, will give good performance in situations where a triband Yagi will not."
Shorter boom
William Orr's book shows a 10-15-20 meter, 2-element Quad with boom length of 6′10″.
Internally stackable
Interaction between antennas of a multiband quad are quite low, even when fed with a single feed line.
According to K0SR the false claim that quads "open the band earlier", which suggests that they exhibit a lower angle of radiation than Yagis, has persisted for 50 years computer models agree. He posits that the vertical sides of each element actually radiate the low angle component.
Disadvantages compared to other antennas
Bandwidth
If tuned for maximum gain, the bandwidth for a 3-element quad antenna is limited: Deviation from the design frequency will unbalance the near-resonance condition of the parasitic elements. However, lengthening the director elements, thereby sacrificing approximately 1 dB gain, allows for much broader bandwidth.
Maintenance
A quad is a 3-dimensional antenna so maintenance can be difficult. Even with a tilt-over tower, tall ladders or a bucket truck may be needed. There are devices that will allow the tilting of the tower to the ground to work on a cubical quad antenna, rotator, or tower. It works by letting the quad loops swivel out of the way. When the tower is in the operational position the elements are locked into position.