Recently a Complexity-Action (CA) duality conjecture has been proposed, which relates the quantum complexity of a holographic boundary state to the action of a Wheeler-DeWitt (WDW) patch in the anti-de Sitter (AdS) bulk. In this paper we further investigate the duality conjecture for stationary AdS black holes and derive some exact results for the growth rate of action within the Wheeler-DeWitt (WDW) patch at late time approximation, which is supposed to be dual to the growth rate of quantum complexity of holographic state. Based on the results from the general D-dimensional Reissner-Nordström (RN)-AdS black hole, rotating/charged Bañados-Teitelboim-Zanelli (BTZ) black hole, Kerr-AdS black hole and charged Gauss-Bonnet-AdS black hole, we present a universal formula for the action growth expressed in terms of some thermodynamical quantities associated with the outer and inner horizons of the AdS black holes. And we leave the conjecture unchanged that the stationary AdS black hole in Einstein gravity is the fastest computer in nature.

The direct detection of gravitational wave by Laser Interferometer Gravitational-Wave Observatory indicates the coming of the era of gravitational-wave astronomy and gravitational-wave cosmology. It is expected that more and more gravitational-wave events will be detected by currently existing and planned gravitational-wave detectors. The gravitational waves open a new window to explore the Universe and various mysteries will be disclosed through the gravitational-wave detection, combined with other cosmological probes. The gravitational-wave physics is not only related to gravitation theory, but also is closely tied to fundamental physics, cosmology and astrophysics. In this review article, three kinds of sources of gravitational waves and relevant physics will be discussed, namely gravitational waves produced during the inflation and preheating phases of the Universe, the gravitational waves produced during the first-order phase transition as the Universe cools down and the gravitational waves from the three phases: inspiral, merger and ringdown of a compact binary system, respectively. We will also discuss the gravitational waves as a standard siren to explore the evolution of the Universe.

A novel mechanism for the primordial black hole (PBH) production is proposed as a natural and inevitable consequence of general first-order phase transitions without reference to specific underlying particle physics models. We obtain mutual predictions and constraints between primordial black holes and gravitational waves from phase transitions in the general case. For particular interest, our PBHs generated during a PeV-scale phase transition could make up all the dark matter, while PBHs generated during a MeV-scale phase transition could simultaneously account for LIGO-Virgo coalescence events and NANOGrav 12.5-yr result for the corresponding gravitational waves.