The toolbox to study the Universe grew on 14 September 2015 when the LIGO–Virgo collaboration heard a signal from two colliding black holes between 30 and 250 Hz. Since then, many more gravitational waves have been detected as detectors continue to increase sensitivity. However, the current and future interferometric detectors will never be able to detect gravitational waves below a few Hz due to oceanic activity on Earth. An interferometric space mission, the laser interferometer space antenna, will operate between 1 mHz and 0.1 Hz, leaving a gap in the decihertz band. To detect gravitational-wave signals also between 0.1 and 1 Hz, the Lunar Gravitational-wave Antenna will use an array of seismic stations. The seismic array will be deployed in a permanently shadowed crater on the lunar south pole, which provides stable ambient temperatures below 40 K. A cryogenic superconducting inertial sensor is under development that aims for fm/√Hz sensitivity or better down to several hundred mHz, and thermal noise limited below that value. Given the 106 m size of the Moon, strain sensitivities below 10−20 1/√Hz can be achieved. The additional cooling is proposed depending on the used superconductor technology. The inertial sensors in the seismic stations aim to make a differential measurement between the elastic response of the Moon and the inertial sensor proof-mass motion induced by gravitational waves. Here, we describe the current state of research toward the inertial sensor, its applications, and additional auxiliary technologies in the payload of the lunar gravitational-wave detection mission.