The Physics arXiv Blog has a really interesting article about the connection between quantum theory and the question of whether or not P = NP. The article starts with the question of why we can't observe quantum phenomena at the macroscopic scale. Arkady Bolotin of Ben-Gurion University says that the key to understanding the problem is to treat it as a problem of computational complexity.
Briefly—see the article for details—the idea is that while it is possible to solve Schrödinger’s equation for simple systems at the quantum scale, it becomes increasingly difficult as the number of particles increases. Bolotin asks what if it's impossible to solve Schrödinger's equation for macroscopic systems in reasonable time. He shows that it is indeed impossible providing P ≠ NP.
The article outlines the argument in terms accessible to those not expert in quantum mechanics. To my—mostly uninformed—mind this provides a persuasive argument for the notion that P ≠ NP. It's not a proof of course but it is persuasive. If you have the slightest interest in the P ≠ NP problem or are fascinated by the counter intuitive results of quantum mechanics, I'm sure you'll enjoy the article.