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In the past a few decades, a variety of fields-from information theory to economic game theory to cryptography-have explored what happens when quantum decisions are allowed in classical frameworks or as a computational mechanism to improve classical algorithms. 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 strategical 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 combinatorial games whose quantum extensions are intractable?• Quantum Collapses in Complexity: Are there PSPACE-complete combinatorial 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? * Supported by the Simons Investigator Award for fundamental & curiosity-driven research and NSF grant CCF-1815254.

We settle two long-standing complexity-theoretical questions-open since 1981 and 1993-in combinatorial game theory (CGT).We prove that the Grundy value (a.k.a. nim-value, or nimber) of Undirected Geography is PSPACE-complete to compute. This exhibits a stark contrast with a result from 1993 that Undirected Geography is polynomial-time solvable. By distilling to a simple reduction, our proof further establishes a dichotomy theorem, providing a "phase transition to intractability" in Grundy-value computation, sharply characterized by a maximum degree of four: The Grundy value of Undirected Geography over any degree-three graph is polynomial-time computable, but over degree-four graphs-even when planar & bipartite-is PSPACE-hard. Additionally, we show, for the first time, how to construct Undirected Geography instances with Grundy value * n and size polynomial in n.We strengthen a result from 1981 showing that sums of tractable partisan games are PSPACEcomplete in two fundamental ways. First, since Undirected Geography is an impartial ruleset, we extend the hardness of sums to impartial games, a strict subset of partisan. Second, the 1981 construction is not built from a natural ruleset, instead using a long sum of tailored short-depth game positions. We use the sum of two Undirected Geography positions to create our hard instances. Our result also has computational implications to Sprague-Grundy Theory (1930s) which shows that the Grundy value of the disjunctive sum of any two impartial games can be computed-in polynomial time-from their Grundy values. In contrast, we prove that assuming PSPACE = P, there is no general polynomial-time method to summarize two polynomial-time solvable impartial games to efficiently solve their disjunctive sum.