Polar codes have emerged as a promising error-correcting code due to channel-achieving error-correcting performance and simple decoder structure. The successive cancellation list decoding provides the best error-correction performance with practical code lengths, but it suffers from the large area occupation that increases with the list size and code length, especially due to large memory requirements. In this paper, an area-efficient SCL polar decoder is proposed using a quaternary-tree architecture. We propose three different architectures for decoding unit, path memory and pointer memory that aim at reducing the memory area and flip-flop updates. The quaternary-tree architecture reduces the internal LLR memory area by combining the basic operations of two stages into one. The previous path memory architecture is modified to reduce the crossbar area and path updates. In addition, the pointer memory architecture is introduced to support the critical pointer first copy method that reduces pointer updates. The implementation results show that the proposed methods can save up to 57% total memory area and thus 41% total decoder area compared with baseline SCL decoder. The proposed quaternary-tree based SCL decoder yields the lower area occupation than the state-of-the-art SCL decoders.