From Quora: How does evolution explain the quantum gap in intelligence between humans and all other species?

By a "quantum gap" I imagine that you mean a large qualitative difference in intelligence that needs to be explained. But the differences we see are arguably a matter of degree (very large degree, but still).

A century ago, asked what separates humans from other animals, we might have said language,  tool use, symbolic reasoning, reasoning with probabilities, analogical reasoning, reasoning about the future, cultural transmission of behaviors, and so forth. But as we've learned more about animals' abilities, the gaps become smaller.

For example, human languages are much more sophisticated than any animal language. One of the most striking characteristic of a human language is its generativity, something that for a long time was thought to be impossible for animals to manage. But that's not necessarily the case [Gentner et al., 2006]:

European starlings (Sturnus vulgaris) accurately recognize acoustic patterns defined by a recursive, self-embedding, context-free grammar. They are also able to classify new patterns defined by the grammar and reliably exclude agrammatical patterns. Thus, the capacity to classify sequences from recursive, centre-embedded grammars is not uniquely human.

Or take tool use, which was once thought of as a hallmark of human intelligence. When we started observing animals using tools (even animals as simple as wasps), the criterion was changed to making rather than only using tools. And when animals such as chimpanzees and crows were observed making tools, the criterion was narrowed further--using tools to make other tools. (Other tool-related capabilities have been suggested as well.) But chimpanzees can be taught to make stone tools by knapping [Roffman et al., 2006]:

Here we describe the ability of two language-competent bonobos (10), Kanzi (KZ; male, age 30 y) and Pan-Banisha (PB; female, age 28 y), to produce novel stone tools and effectively use them, supporting the hypothesis that present-day Pan exhibit technological competencies formerly assigned only to the Homo genus. In the 1990s, KZ and PB were taught by Toth et al. (11, 12) to knap flint flakes and use their sharp edges to cut rope or leather.

Or reasoning about the future [Clayton et al., 2003]:

A more convincing case of planning was provided by Osvath and Osvath. In a recent series of experiments, these authors demonstrated that when selecting a tool for use in the future, chimpanzees and orangutans can override immediate drives in favor of future needs. One of the most striking examples of the spoon test in animals comes from recent studies of the food-caching scrub-jays. In the laboratory, work by Raby and colleagues showed that our jays can spontaneously plan for tomorrow’s breakfast without reference to their current motivational state.

You start to see a pattern: Someone speculates, "This may be a uniquely human intellectual capacity." And then scientists in animal cognition decide to test whether that's true...

The overall point is that what looks like a huge qualitative difference may not be; it could be something more like me in a race against Usain Bolt. Evolution can explain that.