How detailed should the project plan be in an early stage startup?

Depends on what you mean by a detailed plan.

To raise an institutional seed round, you need a solid plan for the first 12 months and a beacon to the 3 year mark. In particular, the first 3-6 months will be under scrutiny. You will need to demonstrate you have thought through how you are going to validate your hypotheses, develop your products, acquire your customers and so forth in adequate detail that proves you are working from field data and not from a work of science fiction.

In general we are not fond of Gantt charts for the early stages of a startup. That said, it is an easy way to visualize timeline, milestones and deliverables, and many find this visualization helpful. What might be helpful in that case is for you to think about the major milestones and deliverables in the next 6-12 months and then lay it out on the Gantt Chart. The chart should not have more than 10 or 20 lines. If you have a plan at this level, it will help you tie your daily, weekly and monthly progress to a bigger milestone and will help you gauge whether you are on t rack. 

Lastly, we are also not fond of a centralized planning process in which one person does all of the planning. Instead, the high level milestones should be agreed upon and ratified by the founding team, and then each workstream should be planned by the person in charge of that workstream. This makes sure the plan incorporates the best thinking from the people doing the work, and also gives each person agency and ownership of their own workstream.