This paper explores the quantum generalizations of classical Markov Chain Monte Carlo (MCMC) methods, focusing on the Glauber and Metropolis dynamics. The authors propose efficient quantum implementations of discrete-time MCMC algorithms that satisfy detailed balance, a key property ensuring convergence to the target distribution. They introduce a new construction for Metropolis sampling and an alternative highly coherent quantum generalization of detailed balanced dynamics, which resembles a physically derived master equation. The paper also studies the properties of these constructions, including their uniqueness, locality, and energy uncertainty. Additionally, it provides a systematic approach to quantum generalizations of classical MCMC methods and discusses the potential of these constructions in various applications. The authors highlight the connection between MCMC methods and physical thermodynamic models of open quantum systems, emphasizing the importance of detailed balance in both classical and quantum settings. The paper concludes with a discussion on open problems and future directions, including the mixing times of these constructions and their practical implications.This paper explores the quantum generalizations of classical Markov Chain Monte Carlo (MCMC) methods, focusing on the Glauber and Metropolis dynamics. The authors propose efficient quantum implementations of discrete-time MCMC algorithms that satisfy detailed balance, a key property ensuring convergence to the target distribution. They introduce a new construction for Metropolis sampling and an alternative highly coherent quantum generalization of detailed balanced dynamics, which resembles a physically derived master equation. The paper also studies the properties of these constructions, including their uniqueness, locality, and energy uncertainty. Additionally, it provides a systematic approach to quantum generalizations of classical MCMC methods and discusses the potential of these constructions in various applications. The authors highlight the connection between MCMC methods and physical thermodynamic models of open quantum systems, emphasizing the importance of detailed balance in both classical and quantum settings. The paper concludes with a discussion on open problems and future directions, including the mixing times of these constructions and their practical implications.