There are several challenges and constraints that prevent us from finding a consistent and complete theory of quantum gravity. Some of them are:

• Conceptual: We do not have a clear understanding of the nature of spacetime and matter at the quantum level, and how they interact with each other. We also do not know how to reconcile the apparent conflict between the deterministic nature of general relativity and the probabilistic nature of quantum mechanics. We do not have a satisfactory resolution of the measurement problem and the quantum state collapse, especially in the context of gravity.
• Mathematical: We do not have a well-defined mathematical framework that can accommodate both quantum mechanics and general relativity in a consistent way. We also do not have a rigorous way of dealing with the infinities and singularities that arise in both theories when applied to extreme situations.
• Physical: We do not have a clear physical picture of what quantum gravity phenomena look like, and what are the observable consequences of different approaches to quantum gravity. We also do not have a clear criterion for choosing between different approaches, or for testing their predictions.
• Experimental: We do not have access to the regimes where quantum gravity effects are expected to be significant, such as near the Planck scale or inside black holes. The current experimental data are insufficient to distinguish between different approaches or to falsify any of them.

Different approaches to quantum gravity try to deal with these challenges and constraints in different ways. Some of them are:

• String theory: This is an approach that tries to unify all fundamental interactions, including gravity, into a single framework based on one-dimensional objects called strings. It avoids the problem of quantizing gravity by replacing point-like particles with strings, which have finite size and can vibrate in different modes. It also avoids some of the infinities and singularities by introducing extra dimensions of space and dualities between different descriptions.
• Loop quantum gravity: This is an approach that tries to quantize gravity directly, without unifying it with other forces. It uses a discrete representation of spacetime based on loops and networks, which are invariant under diffeomorphisms. It also avoids some of the infinities and singularities by introducing a minimal length scale and a discrete spectrum of geometric quantities.
• Other approaches: There are many other approaches to quantum gravity, such as causal dynamical triangulations, causal sets, asymptotic safety, non-commutative geometry, emergent gravity, etc. Each of them has its own advantages and disadvantages, and its own way of addressing the challenges and constraints mentioned above.

Some possible ways to overcome these challenges or test these approaches are:

• Theoretical developments: There is still much work to be done on developing the theoretical foundations, mathematical tools, physical interpretations, and conceptual clarifications of different approaches to quantum gravity. There is also much scope for finding connections, similarities, and differences between different approaches, and for exploring new ideas and directions.
• Phenomenological studies: There is still much room for exploring the possible observable consequences of different approaches to quantum gravity, such as deviations from classical general relativity or standard quantum mechanics, signatures of quantum spacetime structure or topology change, effects on cosmology or astrophysics, etc. There is also much need for finding ways to constrain or test these predictions using existing or future experiments or observations

Source :

Quantum gravity - Wikipedia. https://en.wikipedia.org/wiki/Quantum_gravity.

Conceptual and Technical Challenges of Quantum Gravity. https://link.springer.com/article/10.1007/s10773-020-04520-2.

Frontiers of Quantum Gravity: shared challenges, converging directions. https://arxiv.org/pdf/2207.10618.pdf.

Frontiers of Quantum Gravity: shared challenges .... https://arxiv.org/abs/2207.10618.

An introduction to quantum gravity. https://arxiv.org/abs/1108.3269.

Quantum Gravity - Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/quantum-gravity/.