We study efficient algorithms for the Euclidean (k)-Center problem, focusing on the regime of large (k). We take the approach of data reduction by considering (\alpha)-coreset, which is a small subset (S) of the dataset (P) such that any (\beta)-approximation on (S) is an ((\alpha + \beta))-approximation on (P). We give efficient algorithms to construct coresets whose size is (k \cdot o(n)), which immediately speeds up existing approximation algorithms. Notably, we obtain a near-linear time (O(1))-approximation when (k = n^c) for any (0 < c < 1). We validate the performance of our coresets on real-world datasets with large (k), and we observe that the coreset speeds up the well-known Gonzalez algorithm by up to (4) times, while still achieving similar clustering cost. Technically, one of our coreset results is based on a new efficient construction of consistent hashing with competitive parameters. This general tool may be of independent interest for algorithm design in high dimensional Euclidean spaces.