In conventional spin glasses, the magnetic interaction is not strongly anisotropic and the entire spin system freezes at low temperature. In La2Cu0.94Li0.06O4, for which the in-plane exchange interaction dominates the interplane one, only a fraction of spins with antiferromagnetic correlations extending to neighboring planes become spin-glass. The remaining spins with only in-plane antiferromagnetic correlations remain spin-liquid at low temperature. Such a novel partial spin freezing out of a spinliquid observed in this cold neutron scattering study is likely due to a delicate balance between disorder and quantum fluctuations in the quasi-two dimensional S=1/2 Heisenberg system.The parent compound for high transition-temperature superconductors, La 2 CuO 4 , is an antiferromagnetic (AF) insulator. Magnetic exchange interaction J between the nearest neighbor S=1/2 spins of Cu 2+ ions in the CuO 2 plane is several orders of magnitude stronger than the interplane exchange interaction, making quantum spin fluctuations an essential ingredient for magnetic properties in the quasi-two-dimensional (2D) Heisenberg system [1,2,3]. The Néel temperature T N of La 2 CuO 4 is suppressed rapidly by 2−3% hole dopants such as Sr, Ba or Li [4,5], while it is suppressed with isovalent Cu substitution at a much higher concentration close to the site dilution percolating threshold [6]. The strong effect of holes has been shown to be related to induced magnetic vortices, which are topological defects in 2D systems [7]. The paramagnetic phase exposed by hole doping at T ≪ J/k B is dominated by zero-point quantum spin fluctuations and is referred to as a quantum spin liquid [1]. The approach of the Néel transition toward zero temperature promises detailed predictions for spin dynamics by extending the celebrated theory of critical phenomena for classical statistical systems to quantum statistical systems [1,8].However, in a wide doping range of La 2 Cu 1−x Li x O 4 below ∼10 K, a spin-glass transition has been reported in muon spin rotation (µSR) [5], nuclear quadrupole resonance (NQR)[9] and magnetization[10] studies. A similar magnetic phase diagram has also been reported for La 2−x Sr x CuO 4 and Y 1−x Ca x Ba 2 Cu 3 O 6 [11,12,13]. In conventional spin glasses, magnetic interactions are more or less isotropic in space, and the whole spin system is believed to be frozen in the spin-glass phase [14]. If the spin-glass phase in hole-doped cuprates behaved as in conventional spin-glasses, the ground state would be a spin-glass instead of Néel order or quantum spin liquid, and as pointed out by Hasselmann et al. [15], the quantum critical point of the AF phase would be preempted.Recently, 2D spin fluctuations in La 2 Cu 1−x Li x O 4 (0.04 ≤ x ≤ 0.1) were observed to remain liquid-like below the spin-glass transition temperature T g [16,17]. The characteristic energy of 2D spin fluctuations saturates at a finite value below ∼50 K as expected for a quantum spin liquid[1], instead of becoming zero at T g as for spin-glass materials [14,18...