This paper considers the distributed optimization problem where each node of a peer-to-peer network minimizes a finite sum of objective functions by communicating with its neighboring nodes. In sharp contrast to the existing literature where the fastest distributed algorithms converge either with a global linear or a local superlinear rate, we propose a distributed adaptive Newton (DAN) algorithm with a global quadratic convergence rate. Our key idea lies in the design of a finite-time set-consensus method with Polyak's adaptive stepsize. Moreover, we introduce a low-rank matrix approximation (LA) technique to compress the innovation of Hessian matrix so that each node only needs to transmit message of dimension $\mathcal{O}(p)$ (where $p$ is the dimension of decision vectors) per iteration, which is essentially the same as that of first-order methods. Nevertheless, the resulting DAN-LA converges to an optimal solution with a global superlinear rate. Numerical experiments on logistic regression problems are conducted to validate their advantages over existing methods.