Recently, a standardized framework was proposed for introducing quantum-inspired moves in mathematical games with perfect information and no chance. The beauty of quantum games-succinct in representation, rich in structures, explosive in complexity, dazzling for visualization, and sophisticated for strategic reasoning-has drawn us to play concrete games full of subtleties and to characterize abstract properties pertinent to complexity consequence. Going beyond individual games, we explore the tractability of quantum combinatorial games as whole, and address fundamental questions including: Quantum Leap in Complexity: Are there polynomial-time solvable games whose quantum extensions are intractable? Quantum Collapses in Complexity: Are there PSPACE-complete games whose quantum extensions fall to the lower levels of the polynomial-time hierarchy? Quantumness Matters: How do outcome classes and strategies change under quantum moves? Under what conditions doesn't quantumness matter? PSPACE Barrier for Quantum Leap: Can quantum moves launch PSPACE games into outer polynomial space We show that quantum moves not only enrich the game structure, but also impact their computational complexity. In settling some of these basic questions, we characterize both the powers and limitations of quantum moves as well as the superposition of game configurations that they create. Our constructive proofs-both on the leap of complexity in concrete Quantum Nim and Quantum Undirected Geography and on the continuous collapses, in the quantum setting, of complexity in abstract PSPACE-complete games to each level of the polynomial-time hierarchy-illustrate the striking computational landscape over quantum games and highlight surprising turns with unexpected quantum impact. Our studies also enable us to identify several elegant open questions fundamental to quantum combinatorial game theory (QCGT).

中文翻译：

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量子组合博弈：结构和计算复杂性

最近，提出了一个标准化框架，用于在具有完美信息且没有机会的数学游戏中引入受量子启发的移动。量子游戏的美——表现形式简洁、结构丰富、复杂性具有爆炸性、可视化令人眼花缭乱、战略推理复杂——吸引我们玩充满微妙的具体游戏，并表征与复杂性结果相关的抽象属性。超越单个游戏，我们从整体上探索量子组合游戏的易处理性，并解决基本问题，包括： 复杂性中的量子飞跃：是否存在量子扩展难以处理的多项式时间可解游戏？复杂性中的量子崩溃：是否存在量子扩展落入多项式时间层次结构的较低级别的 PSPACE 完全游戏？量子问题：结果类别和策略如何在量子移动下发生变化？在什么条件下量子性不重要？量子跃迁的 PSPACE 障碍：量子移动能否将 PSPACE 游戏发射到外部多项式空间我们证明量子移动不仅丰富了游戏结构，而且影响了它们的计算复杂度。在解决其中一些基本问题时，我们描述了量子运动的力量和局限性，以及它们创造的游戏配置的叠加。我们的建设性证明——无论是关于具体 Quantum Nim 和 Quantum Undirected Geography 的复杂性飞跃，还是关于量子环境中的连续崩溃，抽象 PSPACE 完全游戏的复杂性到多项式时间层次结构的每个级别 - 说明了量子游戏中引人注目的计算景观，并突出了具有意外量子影响的令人惊讶的转变。我们的研究还使我们能够确定量子组合博弈论 (QCGT) 的几个基本的优雅开放问题。