Core Concepts
The competition complexity of correlated prophet inequalities depends on the number of original rewards, and blockthreshold algorithms may require an infinite number of additional rewards when correlations are present. The authors develop asymptotically optimal algorithms for different arrival models and show that the competition complexity exhibits different dependencies on the number of original rewards.
Abstract
The authors study the competition complexity of the correlated prophet inequality problem, where a decisionmaker observes a sequence of rewards online and must select one in an online fashion. The goal is to design algorithms that can approximate the expected value of the prophet (the optimal offline algorithm) using as few additional copies of the original instance as possible.
The key insights and results are:

Structural Insights:
 The competition complexity of correlated prophet inequalities depends on the number of original rewards n, unlike the independent case.
 Blockthreshold algorithms, which are optimal for the independent case, may require an infinite number of additional rewards when correlations are present.

Quantitative Results:
 For the block arrival model, the authors devise an algorithm with a competition complexity of O(n + log log(1/ε)), which is asymptotically tight.
 For the adversarial arrival model, the authors show a competition complexity of Θ(n/ε), which is also tight.
 The authors also consider the pairwise independent case, where they devise a simplified algorithm with an optimal asymptotic competition complexity of O(log log(1/ε)).
The authors' algorithms are constructive, with upper bounds achieved by implementable efficient algorithms, and their lower bounds are established by explicit hard instances.
Stats
The expected value of the prophet is greater than 1/(1ε) times the expected value of the online algorithm using k copies of the original instance.
The expected value of the prophet is at least 1  1/n times the (11/n)quantile of the maximum value distribution.
Quotes
"Unlike in the independent case, the required number of additional rewards for approximation depends on the number of original rewards, and that blockthreshold algorithms, which are optimal in the independent case, may require an infinite number of additional rewards when correlations are present."
"Our results establish that while the factor n, the number of rewards in the original instance, impacts the competition complexity additively in the block arrival model, it impacts the competition complexity multiplicatively in the adversarial arrival model."