It is commonly accepted that in concentrated solutions or melts high-molecular weight polymers display random-walk conformational properties without long-range correlations between subsequent bonds. This absence of memory means, for instance, that the bond-bond correlation function, $P(s)$, of two bonds separated by $s$ monomers along the chain should exponentially decay with $s$. Presenting numerical results and theoretical arguments for both monodisperse chains and self-assembled (essentially Flory size-distributed) equilibrium polymers we demonstrate that some long-range correlations remain due to self-interactions of the chains caused by the chain connectivity and the incompressibility of the melt. Suggesting a profound analogy with the well-known long-range velocity correlations in liquids we find, for instance, $P(s)$ to decay algebraically as $s^{-3/2}$. Our study suggests a precise method for obtaining the statistical segment length \bstar in a computer experiment.