We present two parallel repetition theorems for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our first theorem shows that for a $k$-player free game $G$ with entangled value $\mathrm{val}^*(G) = 1 - ε$, the $n$-fold repetition of $G$ has entangled value $\mathrm{val}^*(G^{\otimes n})$ at most $(1 - ε^{3/2})^{Ω(n/sk^4)}$, where $s$ is the answer length of any player. In contrast, the best known parallel repetition theorem for the classical value of two-player free games is $\mathrm{val}(G^{\otimes n}) \leq (1 - ε^2)^{Ω(n/s)}$, due to Barak, et al. (RANDOM 2009). This suggests the possibility of a separation between the behavior of entangled and classical free games under parallel repetition. Our second theorem handles the broader class of free games $G$ where the players can output (possibly entangled) quantum states. For such games, the repeated entangled value is upper bounded by $(1 - ε^2)^{Ω(n/sk^2)}$. We also show that the dependence of the exponent on $k$ is necessary: we exhibit a $k$-player free game $G$ and $n \geq 1$ such that $\mathrm{val}^*(G^{\otimes n}) \geq \mathrm{val}^*(G)^{n/k}$. Our analysis exploits the novel connection between communication protocols and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP 2014). We demonstrate that better communication protocols yield better parallel repetition theorems: our first theorem crucially uses a quantum search protocol by Aaronson and Ambainis, which gives a quadratic speed-up for distributed search problems. Finally, our results apply to a broader class of games than were previously considered before; in particular, we obtain the first parallel repetition theorem for entangled games involving more than two players, and for games involving quantum outputs.

Khanna and Sudan [2] studied a natural model of continuous time channels where signals are corrupted by the effects of both noise and delay, and showed that, surprisingly, in some cases both are not enough to prevent such channels from achieving unbounded capacity. Inspired by their work, we consider channels that model continuous time communication with adversarial delay errors. The sender is allowed to subdivide time into an arbitrarily large number M of micro-units in which binary symbols may be sent, but the symbols are subject to unpredictable delays and may interfere with each other. We model interference by having symbols that land in the same micro-unit of time be summed, and we study k-interference channels, which allow receivers to distinguish sums up to the value k. We consider both a channel adversary that has a limit on the maximum number of steps it can delay each symbol, and a more powerful adversary that only has a bound on the average delay. We give precise characterizations of the threshold between finite and infinite capacity depending on the interference behavior and on the type of channel adversary: for max-bounded delay, the threshold is at Dmax= Θ (M log (min{k, M})), and for average bounded delay the threshold is at Davg= Θ (√(M min{k, M})).