# Deep Thinking by Markov Chain of Continuous Thoughts

arXiv: https://arxiv.org/abs/2509.25020 · HTML: https://mlsanity.com/paper/2509.25020/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Placed in Prompting and chain-of-thought reasoning, one of 17 directions in LLM reasoning.
- Builds on 23 of its 32 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## Placed in

- LLM reasoning / [Prompting and chain-of-thought reasoning](https://mlsanity.com/lr/prompting-and-chain-of-thought-reasoning/)

## Builds on (most cited first, 23 of 23)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Qwen2 Technical Report](https://mlsanity.com/paper/2407.10671/) (2024)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Are NLP Models really able to Solve Simple Math Word Problems?](https://mlsanity.com/paper/2103.07191/) (2021)
- [Simple and Effective Masked Diffusion Language Models](https://mlsanity.com/paper/2406.07524/) (2024)
- [Training Large Language Models to Reason in a Continuous Latent Space](https://mlsanity.com/paper/2412.06769/) (2024)
- [Scaling up Test-Time Compute with Latent Reasoning: A Recurrent Depth Approach](https://mlsanity.com/paper/2502.05171/) (2025)
- [Solving General Arithmetic Word Problems](https://mlsanity.com/paper/1608.01413/) (2016)
- [On the Planning Abilities of Large Language Models : A Critical Investigation](https://mlsanity.com/paper/2305.15771/) (2023)
- [Towards Large Reasoning Models: A Survey of Reinforced Reasoning with Large Language Models](https://mlsanity.com/paper/2501.09686/) (2025)
- [To CoT or not to CoT? Chain-of-thought helps mainly on math and symbolic reasoning](https://mlsanity.com/paper/2409.12183/) (2024)
- [Looped Transformers as Programmable Computers](https://mlsanity.com/paper/2301.13196/) (2023)
- [From Explicit CoT to Implicit CoT: Learning to Internalize CoT Step by Step](https://mlsanity.com/paper/2405.14838/) (2024)
- [CODI: Compressing Chain-of-Thought into Continuous Space via Self-Distillation](https://mlsanity.com/paper/2502.21074/) (2025)
- [Implicit Chain of Thought Reasoning via Knowledge Distillation](https://mlsanity.com/paper/2311.01460/) (2023)
- [Likelihood-Based Diffusion Language Models](https://mlsanity.com/paper/2305.18619/) (2023)
- [Looped Transformers are Better at Learning Learning Algorithms](https://mlsanity.com/paper/2311.12424/) (2023)
- [Think Silently, Think Fast: Dynamic Latent Compression of LLM Reasoning Chains](https://mlsanity.com/paper/2505.16552/) (2025)
- [A Survey on Diffusion Language Models](https://mlsanity.com/paper/2508.10875/) (2025)
- [SIM-CoT: Supervised Implicit Chain-of-Thought](https://mlsanity.com/paper/2509.20317/) (2025)
- [Learning Instance-Specific Augmentations by Capturing Local Invariances](https://mlsanity.com/paper/2206.00051/) (2022)
- [Looking Beyond The Top-1: Transformers Determine Top Tokens In Order](https://mlsanity.com/paper/2410.20210/) (2024)

## Built on by (most cited first, 9 of 9)

- [The Latent Space: Foundation, Evolution, Mechanism, Ability, and Outlook](https://mlsanity.com/paper/2604.02029/) (2026)
- [Render-of-Thought: Rendering Textual Chain-of-Thought as Images for Visual Latent Reasoning](https://mlsanity.com/paper/2601.14750/) (2026)
- [Step-Level Sparse Autoencoder for Reasoning Process Interpretation](https://mlsanity.com/paper/2603.03031/) (2026)
- [Unlocking the Black Box of Latent Reasoning: An Interpretability-Guided Approach to Intervention](https://mlsanity.com/paper/2606.01243/) (2026)
- [Markovian Scale Prediction: A New Era of Visual Autoregressive Generation](https://mlsanity.com/paper/2511.23334/) (2025)
- [Inference-Time Rethinking with Latent Thought Vectors for Math Reasoning](https://mlsanity.com/paper/2602.06584/) (2026)
- [Internalized Reasoning for Long-Context Visual Document Understanding](https://mlsanity.com/paper/2604.02371/) (2026)
- [RuPLaR : Efficient Latent Compression of LLM Reasoning Chains with Rule-Based Priors From Multi-Step to One-Step](https://mlsanity.com/paper/2605.09346/) (2026)
- [Next Thoughts Are Distributions: Generative Autoregressive Reasoning in the Latent Space](https://mlsanity.com/paper/2609.33271/) (2026)

## Abstract (arXiv)

Transformer-based models can perform complicated reasoning by generating reasoning paths token by token. While effective, this approach often requires generating thousands of tokens to solve a single problem, which can be slow and computationally expensive. More importantly, it involves a discrete sampling operation at the end of each time step, creating an information bottleneck across time steps. In this work, we propose MarCos, an improvement of the transformer structure that allows fully continuous reasoning at the thought level. Unlike traditional transformer layers, which focus on refining token predictions at each time step, layers in MarCos map a continuous representation of a stepwise thought to the distribution of the next thought. This enables us to achieve multi-step reasoning in a single pass of MarCos. Preliminary experimental results on synthetic and real-world math tasks show the great potential of MarCos. Notably, we observe that the increased information bandwidth of MarCos elicits the ability of parallel thinking, in contrast to single-threaded thinking in traditional transformers. Meanwhile, in real-world math tasks, MarCos achieves more than 10× speedup in wall-clock time with the same level of accuracy. Our code is available at https://github.com/Ljyustc/MarCos.