TITLE: Any important consequences with presupposition of $\mathbf{P} \neq \mathbf{NP}$
QUESTION [19 upvotes]: As we know, there are lots of consequences with the presupposition of the Riemann Hypothesis.
Similarly, are there any important consequences with the presupposition of $\mathbf{P} \neq \mathbf{NP}$ ?
An alternative statement of $\mathbf{P} \neq \mathbf{NP}$ is the extended Church-Turing Thesis. So if we have an speedup algorithm of other model than classic Turing Machine, we have to find an new algorithm for Turing Machine with the assumption $\mathbf{P} \neq \mathbf{NP}$ ? that means we have to find new speedup algorithm of factoring and the like.

REPLY [17 votes]: Let me take a slightly different angle on this question and interpret it as asking what sorts of hardness results require P ≠ NP only, and which ones require stronger assumptions.
By itself, P ≠ NP implies (or more pedantically, is known to imply) less than what many people think.  For example, here are some of the things that P ≠ NP is  not known to imply:

Factoring has no polynomial time algorithm.
Graph isomorphism has no polynomial time algorithm.
One-way functions exist.
There is in general no short certificate that a graph is not Hamiltonian.
There is no family of polynomial size circuits for SAT.
NP-complete problems have no subexponential time algorithms.
NP-complete problems are hard on average.
There is no randomized polynomial time algorithm for an NP-complete problem.
There is no polynomial time algorithm for an NP-complete problem on a quantum computer.

On the other hand, P ≠ NP does imply some results that might seem at first to be stronger than P ≠ NP. A notable class of examples are inapproximability results that follow from the PCP theorem.  For example, if P ≠ NP then there is no polynomial time approximation algorithm for the size of the maximum independent set of a graph that is guaranteed to get within a constant (or even logarithmic) factor of the optimum.  Another example is Mahaney's theorem that if P ≠ NP then there is no sparse set that is NP-complete.