We present an approach to the calculation of arbitrary spectral, thermal and excited state properties within the full configuration interaction quantum Monte Carlo framework. This is achieved via an unbiased projection of the Hamiltonian eigenvalue problem into a space of stochastically sampled Krylov vectors, thus enabling the calculation of real-frequency spectral and thermal properties and avoiding explicit analytic continuation. We use this approach to calculate temperature-dependent properties and one-and two-body spectral functions for various Hubbard models, as well as isolated excited states in ab initio systems.Quantum Monte Carlo (QMC) in its various guises, is undoubtedly one of the most important approaches for accurate elucidation of properties for correlated systems [1][2][3][4][5]. However, these successes have focused primarily on the ground state energy and observables which commute with the Hamiltonian. Critical importance for a deeper understanding of correlated systems comes from dynamic correlation functions and spectral quantities. These mirror how we perceive our environment, namely by perturbing a system and measuring its response -the basis of nearly all spectroscopic and experimental approaches. This gives us direct insight into optical, magnetic and other beyond-ground-state properties, and allow for direct comparison to experimental results.Direct access to dynamic properties is a persistent difficulty for QMC approaches in general. While in the absence of a sign problem, unbiased imaginary-time spectra can be obtained [6][7][8], the analytic continuation to physical, real-frequency functions is notoriously ill-conditioned and can lead to artefacts and smoothing of features.[9] For more general Fermionic systems, higher temperatures must be simulated to alleviate the sign problem[10], while nodal constraints bias towards a particular solution and are difficult to extend to spectra [1,7,11]. Alternatively, projections into effective Hamiltonians have been able to obtain a few low-energy states, but again these are isolated states rather than practical approaches for thermal or spectral quantities [12,13], while a modification of the propagator can lead to debilitating timestep issues [14].Here, we present a new QMC approach for computing dynamic correlation functions, temperature-dependent quantities and isolated excited states for correlated quantum systems, even in the presence of a sign-problem. These correlation functions are unbiased in the limit of large averaging, and exact in the limit of large walker number. This is achieved by extending the recently developed Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method [2,15,16], by combining it with ideas from the dynamical and finite-temperature Lanczos (FTLM) methods. [17][18][19] The key advantage of the approach is that it avoids any explicit storage over the full Hilbert space, instead only storing occupied states in the * nsb37@cam.ac.uk † george.booth@kcl.ac.uk discretized wavefunction at each snapshot. This all...