Sparse signals are encountered in a broad range of applications. In order to process these signals using digital hardware, they must be first sampled and quantized using an analog-to-digital convertor (ADC), which typically operates in a serial scalar manner. In this work, we propose a method for serial quantization of sparse time sequences (SQuaTS) inspired by group testing theory, which is designed to reliably and accurately quantize sparse signals acquired in a sequential manner using serial scalar ADCs. Unlike previously proposed approaches which combine quantization and compressed sensing (CS), our SQuaTS scheme updates its representation on each incoming analog sample and does not require the complete signal to be observed and stored in analog prior to quantization. We characterize the asymptotic tradeoff between accuracy and quantization rate of SQuaTS as well as its computational burden. We also propose a variation of SQuaTS, which trades rate for computational efficiency. Next, we show how SQuaTS can be naturally extended to distributed quantization scenarios, where a set of jointly sparse time sequences are acquired individually and processed jointly. Our numerical results demonstrate that SQuaTS is capable of achieving substantially improved representation accuracy over previous CS-based schemes without requiring the complete set of analog signal samples to be observed prior to its quantization, making it an attractive approach for acquiring sparse time sequences.